January 01, 1970
We introduce a Rayleigh-quotient minimax method for locating maximal one-sided saddle-node bifurcations in nonlinear equations, including non-variational ones. The method avoids branch continuation and instead selects the critical point directly through the minimax bifurcation formula \[\lambda^* := \sup_{u\in\mathcal{U}^o} \inf_{v\in\mathcal{S}\setminus\{0\}} \mathcal{R}(u,v),\] where \(\mathcal{R}\) is a two-variable extended Rayleigh quotient on fixed cones. A saddle point of this quotient simultaneously determines the critical parameter, the bifurcation solution, and the adjoint singularity relation. This gives a direct characterization of the bifurcation threshold and leads to Galerkin minimax approximations, a posteriori parameter-margin estimates, and perturbation bounds for the critical value. The abstract assumptions are verified for nonlinear elliptic systems, including non-potential systems.
Bifurcation ,saddle-node bifurcation ,nonlinear elliptic equations ,variational methods ,finite element method
35P30 ,35J10 ,35J60 ,65N30
Saddle-node bifurcations, also known as folds or turning points, are among the basic mechanisms by which solution branches of nonlinear equations appear or disappear. In nonlinear differential equations, they often mark critical parameter values beyond which solutions in a prescribed class cease to exist. Locating such values is therefore a central problem both in analysis and in numerical computation.
The standard way to detect saddle-node bifurcations is to continue a branch of solutions and monitor the loss of invertibility of the linearized operator. This is the basis of classical local and global bifurcation theory, including the Crandall–Rabinowitz theorem, Rabinowitz’s global theorem, and the bifurcation theory of Krasnosel’skiı̆, as well as of numerical continuation methods [1]–[7]. These methods are powerful, but they are essentially indirect: the critical point is detected through the behaviour of a solution curve. This becomes especially delicate in infinite-dimensional problems, in systems, and in models where positivity or cone constraints are an essential part of the problem.
This paper develops a direct approach to this problem. Maximal one-sided saddle-node points are identified by means of a minimax bifurcation formula. Rather than constructing or following a solution branch, the critical parameter is characterized directly as an extremal value of an extended Rayleigh quotient, so that the minimax formula itself determines the saddle-node value.
Let \(\mathcal{U}^o,\mathcal{S}\subset W\) be prescribed cones in a Banach space \(W\). We consider nonlinear equations of the form \[\label{Gf-intro} \mathcal{F}(u,\lambda) := \mathcal{A}(u)-\lambda\mathcal{G}(u)=0,\quad u\in W,\tag{1}\] where \(\mathcal{A}\) and \(\mathcal{G}\) take values in \(W^*\) and are continuously Fréchet differentiable in a suitable neighbourhood of the admissible set \(\mathcal{U}^o\).
The idea behind the method is simple. If \(u\) is a solution, then testing the equation against \(v\in\mathcal{S}\) gives the extended equation [8] \[\langle\mathcal{A}(u),v\rangle = \lambda \langle\mathcal{G}(u),v\rangle .\] Thus, under the denominator positivity condition imposed below, the parameter \(\lambda\) can be recovered from the extended Rayleigh quotient \[\mathcal{R}(u,v) := \frac{\langle\mathcal{A}(u),v\rangle}{\langle\mathcal{G}(u),v\rangle},\quad \langle\mathcal{G}(u),v\rangle \neq 0.\] We call \(\mathcal{R}\) the extended Rayleigh quotient [9]. Here \(\langle\cdot,\cdot\rangle\) denotes the duality pairing between \(W^*\) and \(W\).
For an admissible solution \(u\), this quotient is independent of \(v\) and equals the corresponding parameter \(\lambda\). This leads to the minimax bifurcation formula \[\label{MainB} \lambda^* := \sup_{u\in\mathcal{U}^o} \inf_{v\in\mathcal{S}\setminus\{0\}} \mathcal{R}(u,v).\tag{2}\] The number \(\lambda^*\) is the natural variational candidate for the maximal parameter value at which admissible solutions may exist. Indeed, every admissible solution satisfies \[\mathcal{R}(u,v)=\lambda \qquad \forall v\in\mathcal{S}\setminus\{0\},\] and therefore \(\lambda\le\lambda^*\). Hence, if \(\lambda^*<+\infty\), equation 1 has no admissible solutions for \(\lambda>\lambda^*\), so that \(\lambda^*\) provides a variational upper threshold for solvability.
The role of the second variable is crucial. The extended quotient does not only provide scalar tests for the parameter; it also carries the dual information needed to identify the point at which the equation becomes singular. Indeed, suppose, formally, that a pair \((u^*,v^*)\) realizes the minimax value and satisfies the corresponding Euler conditions \[D_v\mathcal{R}(u^*,v^*)=0, \qquad D_u\mathcal{R}(u^*,v^*)=0 .\] Then the first condition recovers the equation itself, \[\mathcal{F}(u^*,\lambda^*)=0,\] whereas the second gives the adjoint kernel relation \[\big\langle D_u\mathcal{F}(u^*,\lambda^*)\varphi, v^* \big\rangle=0 \qquad \forall \varphi\in W .\] Thus the minimax construction is designed not merely to detect a critical parameter value, but to select a singular triple \((u^*,v^*,\lambda^*)\), and hence a singular point \((u^*,\lambda^*)\). In this sense, the quotient is genuinely extended: the additional variable is not an auxiliary testing parameter, but the dual variable through which the singularity condition is encoded.
This variational viewpoint also gives a new way to quantify the distance to the threshold. For a known admissible solution \((u_\lambda,\lambda)\), the gap \(\lambda^*-\lambda\) can be estimated directly through the extended Rayleigh quotient \(\mathcal{R}\), without computing the principal eigenvalue of the linearized operator \(D_u\mathcal{F}(u_\lambda,\lambda)\). This provides computable a posteriori certificates in threshold problems such as pull-in instability, voltage collapse, and tipping phenomena, and also yields perturbation bounds for the maximal one-sided saddle-node value.
The paper develops this program in an abstract cone setting. We prove that the saddle-point realization of the minimax formula selects a maximal one-sided saddle-node point, establish the stability of this construction under Galerkin approximation, derive a posteriori parameter-margin certificates and perturbation bounds, and verify the abstract assumptions in model elliptic problems.
The approach builds on earlier work on generalized Rayleigh quotients and inverse variational characterizations [8]–[16]. The novelty of the present paper is the two-cone minimax formulation, its abstract saddle-point interpretation, its stability under Galerkin approximation, and its use in computable certificates for maximal one-sided saddle-node values.
The paper is organized as follows. Section 2 develops the abstract minimax theory. Section 3 records a posteriori and perturbation estimates for the maximal one-sided saddle-node value. Section 4 contains the proofs of the main abstract results. Sections 5 and 6 apply the theory to model elliptic problems. Section 7 contains concluding remarks. The appendices collect the finite-dimensional cone setting, interpolation estimates, and Picone-type inequalities used in the compactness arguments.
Throughout the paper, \(c,C,c_1,C_1,\ldots\) denote positive constants whose values may change from line to line.
Throughout this section, \(W\) denotes a reflexive Banach space and \(\mathcal{C}^0\) an ordered Banach function space. The duality pairing between \(W^*\) and \(W\) is denoted by \(\langle\cdot,\cdot\rangle\), and all order relations are understood in \(\mathcal{C}^0\). We work with a convex cone \(\mathcal{S}^o\subset W\cap\mathcal{C}^0\) and put \[\mathcal{S}:=\overline{\mathcal{S}^o}^{\,\mathcal{C}^0}\cap W .\] The cone is assumed to be generating in the sense that \(\overline{\operatorname{span}\mathcal{S}^o}^{\,W}=W .\)
Finally, we fix a subclass \[\mathcal{U}^o\subset\mathcal{S}^o,\] which will serve as the admissible class of regular solutions.
Finite-dimensional approximations are described by subspaces \(W_r\subset W\), \(N_r:=\dim W_r\), such that \(\bigcup_{r\ge1}W_r\) is dense in \(W\), together with interpolation operators \(\mathcal{I}_r:\mathcal{C}^0\to W_r\). We assume the following compatibility condition for the Galerkin cones.
The Galerkin cones satisfy the following conditions.
For every \(r\ge1\), \(\mathcal{S}_r^o\subset W_r\) is a nonempty open cone, \(\mathcal{S}_r:=\overline{\mathcal{S}_r^o}^{\,W_r}\), and \(\mathcal{S}_r\subset\mathcal{S}\).
The cone \(\mathcal{S}_r\) is polyhedral: there is a basis \(\{\eta_1,\ldots,\eta_{N_r}\}\) of \(W_r\) such that \[\mathcal{S}_r^o = \left\{ v=\sum_{i=1}^{N_r}v^i\eta_i\in W_r: v^i>0,\;i=1,\ldots,N_r \right\}.\]
For every \(u\in\mathcal{U}^o\), the interpolant \(\mathcal{I}_ru\) is well defined and, for all sufficiently large \(r\), \(\mathcal{I}_ru\in\mathcal{S}_r^o\) .
The central object of the abstract framework is the following one-parameter equation on the cone \(\mathcal{S}\): \[\label{Gf} \mathcal{F}(u,\lambda):=\mathcal{A}(u)-\lambda\mathcal{G}(u)=0, \qquad u\in\mathcal{S} .\tag{3}\] Here \(\lambda\in\mathbb{R}\) is the bifurcation parameter, and \(\mathcal{A},\mathcal{G}:W\to W^*\) are continuously Fréchet differentiable on \(\mathcal{U}^o\). Thus, for \(u\in\mathcal{U}^o\), the linearization is well defined and is given by \[D_u\mathcal{F}(u,\lambda)\xi = D_u\mathcal{A}(u)\xi-\lambda D_u\mathcal{G}(u)\xi, \qquad \xi\in W .\]
We impose the following structural assumptions.
If \(u\in\mathcal{S}^o\) weakly satisfies \(\mathcal{F}(u,\lambda)=0\), then \(u\in\mathcal{U}^o\).
\(\langle\mathcal{G}(u),v\rangle>0\) for every \((u,v)\in\mathcal{S}^o\times(\mathcal{S}\setminus\{0\})\).
We shall regard \({\rm (R)}\) and \({\rm (D)}\) as standing assumptions.
Definition 1. A solution \((u_0,\lambda_0)\in\mathcal{U}^o\times\mathbb{R}\) of 3 is called singular* if there exists \(v_0\in\mathcal{S}\setminus\{0\}\) such that \[\bigl\langle D_u\mathcal{F}(u_0,\lambda_0)\xi,v_0\bigr\rangle=0 \qquad \forall \xi\in W .\] It is called a one-sided saddle-node point in \(\mathcal{U}^o\) if there exist \(\varepsilon>0\) and a neighbourhood \(V\) of \(u_0\) in \(W\) such that 3 has no solutions in \(\mathcal{U}^o\cap V\) for \(\lambda\in(\lambda_0,\lambda_0+\varepsilon)\). It is called maximal if every one-sided saddle-node point \((\hat{u},\hat{\lambda})\in\mathcal{U}^o\times\mathbb{R}\) satisfies \(\hat{\lambda}\le\lambda_0\).*
The basic tool of the method is the extended Rayleigh quotient associated with 3 . It is defined by \[\mathcal{R}(u,v):= \frac{\langle\mathcal{A}(u),v\rangle}{\langle\mathcal{G}(u),v\rangle}, \qquad \langle\mathcal{G}(u),v\rangle\neq0 .\] Under \({\rm (D)}\), this quotient is well defined on \(\mathcal{S}^o\times(\mathcal{S}\setminus\{0\})\). For each fixed \(u\in\mathcal{S}^o\), the map \(v\mapsto\mathcal{R}(u,v)\) is continuous on \(\mathcal{S}\setminus\{0\}\) in both the strong and weak topologies of \(W\). Moreover, \(\mathcal{R}\) is continuously Fréchet differentiable on \(\mathcal{U}^o\times\mathcal{S}^o\).
The distinguished variational quantity of the construction is the minimax level \[\label{eq:lambda-star} \lambda^* := \sup_{u\in\mathcal{U}^o} \inf_{v\in\mathcal{S}\setminus\{0\}} \mathcal{R}(u,v).\tag{4}\] It will be shown below that, under the appropriate compactness and consistency assumptions, this value is realized by a singular solution and coincides with the maximal one-sided saddle-node threshold.
Definition 2. The minimax bifurcation formula* is said to hold for 3 with respect to the cone \(\mathcal{S}\) if there exists \((u^*,v^*)\in\mathcal{U}^o\times(\mathcal{S}\setminus\{0\})\) such that \[\label{MBFDefi} \lambda^* = \mathcal{R}(u^*,v^*) = \inf_{v\in\mathcal{S}\setminus\{0\}}\mathcal{R}(u^*,v) = \sup_{u\in\mathcal{U}^o} \inf_{v\in\mathcal{S}\setminus\{0\}} \mathcal{R}(u,v).\tag{5}\] *
A pair \((u_0,v_0)\in\mathcal{U}^o\times(\mathcal{S}\setminus\{0\})\) is called a saddle-point solution pair at the level \(\lambda_0\) if \[\mathcal{F}(u_0,\lambda_0)=0 \quad\text{and}\quad \bigl\langle D_u\mathcal{F}(u_0,\lambda_0)\xi,v_0\bigr\rangle=0 \qquad \forall \xi\in W .\] Thus, when such a pair realizes the minimax level \(\lambda^*<+\infty\), it also provides the singularity relation associated with the corresponding one-sided saddle-node point.
We next pass to Galerkin approximations. The Galerkin approximation of 3 is to find \((u,\lambda)\in\mathcal{S}_r^o\times\mathbb{R}\) such that \[\label{eq:AppEq} \langle\mathcal{F}(u,\lambda),\zeta\rangle=0 \qquad \forall \zeta\in W_r .\tag{6}\] For \(u\in\mathcal{S}_r^o\), set \[\lambda_r(u):= \inf_{v\in\mathcal{S}_r\setminus\{0\}}\mathcal{R}(u,v), \qquad \lambda_r^*:= \sup_{u\in\mathcal{S}_r^o}\lambda_r(u).\]
We use the following quotient-compatibility condition on the Galerkin spaces.
For every \(r\ge1\), \[\langle\mathcal{G}(u),v\rangle>0 \qquad \forall (u,v)\in \mathcal{S}_r^o\times(\mathcal{S}_r\setminus\{0\}).\] Moreover, the finite-dimensional restrictions needed below are \(C^1\) on \(\mathcal{S}_r^o\); in particular, \(\mathcal{R}\) is \(C^1\) on \(\mathcal{S}_r^o\times\mathcal{S}_r^o\).
By the continuity of \(\mathcal{F}\), the residual is sequentially closed in the following sense: if \(u_n\to u\) in \(W\), \(\lambda_n\to\lambda\), and \(\zeta_n\to\zeta\) in \(W\), with \(u_n\in\mathcal{S}_{r_n}^o\) and \(\zeta_n\in W_{r_n}\), then \[\langle\mathcal{F}(u_n,\lambda_n),\zeta_n\rangle \to \langle\mathcal{F}(u,\lambda),\zeta\rangle .\] We impose the following closedness assumption for the linearization.
For every \(u\in\mathcal{U}^o\), \(v\in\mathcal{S}\), \(\xi\in W\), and \(\lambda\in\mathbb{R}\), the following holds: whenever \(u_n\in\mathcal{S}_{r_n}^o\), \(v_n\in\mathcal{S}_{r_n}^o\), \(\xi_n\in W_{r_n}\), \(r_n\to\infty\), and \(\lambda_n\to\lambda\), with \[u_n\to u \quad\text{in }W,\qquad \xi_n\to\xi \quad\text{in }W,\qquad v_n\rightharpoonup v \quad\text{weakly in }W,\] one has \[\bigl\langle D_u\mathcal{F}(u_n,\lambda_n)\xi_n,v_n \bigr\rangle \to \bigl\langle D_u\mathcal{F}(u,\lambda)\xi,v \bigr\rangle .\]
Condition \({\rm (CD)}\) is automatic if \(\mathcal{F}\) is continuously Fréchet differentiable with respect to \(u\) on \(\mathcal{U}^o\times\mathbb{R}\), with values in \(\mathcal{L}(W,W^*)\). Thus \({\rm (CD)}\) only has to be checked separately in applications where the derivative is singular, or where differentiability is available only in a relative sense along admissible Galerkin sequences.
We shall call a Galerkin scheme \[\{W_r,\mathcal{S}_r^o,\mathcal{I}_r\}_{r\ge1}\] admissible if it satisfies \({\rm (C)}\), \({\rm (CQ)}\), and \({\rm (CD)}\).
We now add the assumptions specific to the nonlinear bifurcation problem.
Let \(\{W_r,\mathcal{S}_r^o,\mathcal{I}_r\}_{r\ge1}\) be a fixed admissible Galerkin scheme. We assume the following compactness and boundary-exclusion condition.
For all sufficiently large \(r\ge1\), the following assertions hold.
If \(\lambda_r^*<+\infty\), then there exists \(\alpha_r<\lambda_r^*\) such that \[K_{r,\alpha_r}:= \{u\in\mathcal{S}_r^o:\lambda_r(u)\ge\alpha_r\}\] has compact closure in \(W_r\), and this closure is contained in \(\mathcal{S}_r^o\).
If \(u\in\mathcal{S}_r^o\), \(v\in\mathcal{S}_r\setminus\{0\}\), \[\mathcal{R}(u,v)=\lambda_r^*<+\infty, \qquad D_u\mathcal{R}(u,v)(\xi)=0\quad\forall\xi\in W_r,\] then \(v\in\mathcal{S}_r^o\).
Although condition \({\rm (H)}\) is formulated in terms of a chosen finite-dimensional approximation, its meaning is simple. Assumption \({\rm (h1)}\) says that the relevant superlevel sets remain compactly inside the positive Galerkin cone. Assumption \({\rm (h2)}\) excludes boundary minimizing directions; in applications this is precisely the positivity property expected from the maximum principle for the corresponding linearized Galerkin problem. Thus, for the standard finite-element Galerkin cones used below, these requirements become transparent and are verified by compactness, comparison, and maximum-principle arguments.
Extended Palais–Smale condition. We shall use the notation \((PS)_e\), where the subscript \(e\) stands for “extended”. This indicates that the compactness requirement is not the classical Palais–Smale condition for a one-variable functional, but its two-variable Galerkin analogue adapted to the extended Rayleigh quotient.
Definition 3. The Rayleigh quotient \(\mathcal{R}\) is said to satisfy the extended \((PS)_e\)-condition at level \(\lambda\in\mathbb{R}\) if every sequence \(r_n\to\infty\) and every sequence \((u_n,v_n)\in\mathcal{S}_{r_n}^o\times\mathcal{S}_{r_n}^o\) satisfying \[\label{PSEcond} \begin{cases} \mathcal{R}(u_n,v_n)\to\lambda,\\ D_u\mathcal{R}(u_n,v_n)(\xi)=0 \quad \forall \xi\in W_{r_n},\\ D_v\mathcal{R}(u_n,v_n)(\zeta)=0 \quad \forall \zeta\in W_{r_n}, \end{cases}\qquad{(1)}\] has a subsequence and positive numbers \(t_n>0\) such that, after replacing \(v_n\) by \(t_nv_n\), \[u_n\to\bar u \quad\text{strongly in }W, \qquad v_n\rightharpoonup\bar v \quad\text{weakly in }W,\] with \(\bar u\in\mathcal{S}^o\) and \(\bar v\in\mathcal{S}\setminus\{0\}\). Such a sequence is called an extended \((PS)_e\)-sequence at level \(\lambda\).
Singular Galerkin limits. The first result shows how the compactness encoded in the extended \((PS)_e\)-condition turns finite-dimensional minimax pairs into genuine singular solutions of the continuous problem.
Theorem 4 (Singular limit theorem). Assume that \(\{W_r,\mathcal{S}_r^o,\mathcal{I}_r\}_{r\ge1}\) is an admissible Galerkin scheme. Under the standing assumptions, suppose that condition \({\rm (H)}\) holds and that \(0\le\lambda_r^*<+\infty\) for all sufficiently large \(r\). Then the following assertions hold.
For every sufficiently large \(r\), there exist \(u_r^*,v_r^*\in\mathcal{S}_r^o\) such that \[\label{eq:MMth2} \lambda_r^* = \mathcal{R}(u_r^*,v_r^*) = \min_{v\in\mathcal{S}_r^o}\mathcal{R}(u_r^*,v) = \max_{u\in\mathcal{S}_r^o} \min_{v\in\mathcal{S}_r^o}\mathcal{R}(u,v).\qquad{(2)}\] Moreover, \[\langle\mathcal{F}(u_r^*,\lambda_r^*),\zeta\rangle=0 \quad \forall \zeta\in W_r, \qquad \bigl\langle D_u\mathcal{F}(u_r^*,\lambda_r^*)\xi,v_r^* \bigr\rangle=0 \quad \forall \xi\in W_r .\] Thus \((u_r^*,v_r^*)\) is a finite-dimensional saddle-point pair at the level \(\lambda_r^*\).
Assume, in addition, that, along a subsequence, \(\lambda_r^*\to\hat{\lambda}^*\in\mathbb{R}\), and that \(\mathcal{R}\) satisfies the extended \((PS)_e\)-condition at the level \(\hat{\lambda}^*\). Then, up to a further subsequence and a positive rescaling of \(v_r^*\), \[u_r^*\to u^* \quad\text{strongly in }W, \qquad v_r^*\rightharpoonup v^* \quad\text{weakly in }W,\] where \(u^*\in\mathcal{U}^o\) and \(v^*\in\mathcal{S}\setminus\{0\}\). Moreover, \[\mathcal{F}(u^*,\hat{\lambda}^*)=0 \quad\text{in }W^*, \qquad \bigl\langle D_u\mathcal{F}(u^*,\hat{\lambda}^*)\xi,v^* \bigr\rangle=0 \quad \forall \xi\in W .\]
We next relate the Galerkin saddle levels obtained in Theorem 4 to the continuous minimax value. For \(u\in\mathcal{U}^o\), put \[\lambda(u):= \inf_{v\in\mathcal{S}\setminus\{0\}}\mathcal{R}(u,v).\] Then 4 can be written as \[\lambda^*=\sup_{u\in\mathcal{U}^o}\lambda(u).\] To pass from the Galerkin minimax levels to the continuous minimax value, we need the following one-sided consistency property. It is required only near the top level \(\lambda^*\) and says that the interpolation procedure does not decrease the inner minimax value in the limit.
Assume that \(\lambda^*<+\infty\). There exists \(\delta_0>0\) such that, for every \(u\in\mathcal{U}^o\) with \(\lambda(u)\ge\lambda^*-\delta_0\), \[\liminf_{r\to\infty}\lambda_r(\mathcal{I}_r u) \ge \lambda(u).\]
Main abstract theorem. We can now state the main result of this section. It combines the singular Galerkin limits of Theorem 4 with the one-sided consistency condition to obtain the full minimax bifurcation formula with respect to the cone \(\mathcal{S}\), and identifies the resulting value as the maximal one-sided saddle-node threshold.
Theorem 5 (Minimax bifurcation formula). Assume the standing assumptions and \({\rm (LC)}\). Suppose that condition \({\rm (H)}\) holds, that the minimax levels satisfy \(0<c\le\lambda_r^*\le C<+\infty\) for all sufficiently large \(r\), and that \(\mathcal{R}\) satisfies the extended \((PS)_e\)-condition at every level \(\lambda\in[c,C]\).
Then the minimax bifurcation formula holds for 3 . More precisely, there exists \((u^*,v^*)\in\mathcal{U}^o\times(\mathcal{S}\setminus\{0\})\) such that \[\label{MBFThm} c\le \lambda^* = \mathcal{R}(u^*,v^*) = \inf_{v\in\mathcal{S}\setminus\{0\}}\mathcal{R}(u^*,v) = \sup_{u\in\mathcal{U}^o} \inf_{v\in\mathcal{S}\setminus\{0\}} \mathcal{R}(u,v).\qquad{(3)}\] Moreover, \[\mathcal{F}(u^*,\lambda^*)=0 \quad\text{in }W^*, \qquad \bigl\langle D_u\mathcal{F}(u^*,\lambda^*)\xi,v^* \bigr\rangle=0 \quad \forall \xi\in W .\] In addition, 3 has no solution \(u\in\mathcal{U}^o\) for any \(\lambda>\lambda^*\). Consequently, \((u^*,\lambda^*)\) is a maximal one-sided saddle-node point in \(\mathcal{U}^o\).
Finally, the approximating minimax levels converge to the same value: \(\lambda_r^*\to\lambda^*\). There exist a subsequence, still denoted by \(r\), and pairs \((u_r^*,v_r^*)\in\mathcal{S}_r^o\times\mathcal{S}_r^o\) such that, after a positive rescaling of \(v_r^*\), \[u_r^*\to u^* \quad\text{strongly in }W, \qquad v_r^*\rightharpoonup v^* \quad\text{weakly in }W .\]
Remark 6 (On the verification of the abstract assumptions). The compactness and consistency assumptions used above are not intended to be merely formal hypotheses. Their role is to isolate, in the abstract setting, the analytic mechanisms needed for the minimax construction to select a genuine singular solution.
In the applications below, these assumptions are verified for concrete elliptic problems and admissible finite-dimensional cones. In particular, compactness follows from elliptic estimates and growth conditions, boundary exclusion is obtained from comparison and discrete maximum principles, the extended \((PS)_e\)-condition is obtained using Picone-type inequalities, and the closedness condition \({\rm (CD)}\) is checked separately when the concave term has a singular derivative.
Remark 7 (Dual minimax formula). The orientation in 4 is adapted to maximal one-sided saddle-node values, where admissible solutions cease to exist for larger parameters. The opposite orientation is described by \[\lambda_* := \inf_{u\in\mathcal{U}^o} \sup_{v\in\mathcal{S}\setminus\{0\}} \mathcal{R}(u,v).\] Under the corresponding reversed inequalities and compactness assumptions, the same argument yields minimal one-sided saddle-node values, where admissible solutions cease to exist below the threshold. We omit the details, since they are completely parallel.
The minimax bifurcation formula also gives quantitative information about the maximal one-sided saddle-node value. This section records two simple consequences. First, the extended Rayleigh quotient provides computable a posteriori intervals for \(\lambda^*\), without computing the principal eigenvalue of the linearized operator \[D_u\mathcal{F}(u_\lambda,\lambda).\] Second, the same quotient gives one-sided bounds on the change of \(\lambda^*\) under perturbations of the operator.
Set \[\lambda(u):= \inf_{v\in\mathcal{S}\setminus\{0\}}\mathcal{R}(u,v), \qquad u\in\mathcal{U}^o,\] and \[\Lambda(v):= \sup_{u\in\mathcal{U}^o}\mathcal{R}(u,v), \qquad v\in\mathcal{S}\setminus\{0\}.\] Then \[\lambda^* = \sup_{u\in\mathcal{U}^o}\lambda(u), \qquad \Lambda^* := \inf_{v\in\mathcal{S}\setminus\{0\}}\Lambda(v),\] whenever \(\Lambda^*\) is well defined, and the minimax inequality gives \[\lambda^*\le \Lambda^* .\]
Corollary 8 (A posteriori parameter-margin certificate). Assume that \(\mathcal{R}\) is well defined on \(\mathcal{U}^o\times(\mathcal{S}\setminus\{0\})\). Then, for every \(u\in\mathcal{U}^o\) and every \(v\in\mathcal{S}\setminus\{0\}\) with \(\Lambda(v)<+\infty\), one has \[\label{eq:weak-gap-bound} \lambda(u)\le \lambda^*\le \Lambda(v).\qquad{(4)}\] Equivalently, \[0\le \lambda^*-\lambda(u)\le \Lambda(v)-\lambda(u).\]
In particular, if \(u_\lambda\in\mathcal{U}^o\) is an admissible solution of \(\mathcal{F}(u_\lambda,\lambda)=0\), then, for every \(v\in\mathcal{S}\setminus\{0\}\) with \(\Lambda(v)<+\infty\), \[\label{eq:parameter-margin-certificate} 0\le \lambda^*-\lambda\le \Lambda(v)-\lambda .\qquad{(5)}\] Consequently, if \(\Lambda(v)-\lambda\le\varepsilon\) for some \(v\in\mathcal{S}\setminus\{0\}\), then \[0\le \lambda^*-\lambda\le\varepsilon .\]
Proof. The lower bound in ?? follows directly from the definition of \(\lambda^*\). To prove the upper bound, fix \(v\in\mathcal{S}\setminus\{0\}\). For every \(w\in\mathcal{U}^o\), \[\lambda(w) = \inf_{\zeta\in\mathcal{S}\setminus\{0\}}\mathcal{R}(w,\zeta) \le \mathcal{R}(w,v) \le \sup_{z\in\mathcal{U}^o}\mathcal{R}(z,v) = \Lambda(v).\] Taking the supremum over \(w\in\mathcal{U}^o\), we obtain \(\lambda^*\le\Lambda(v)\).
If \(u_\lambda\) solves \(\mathcal{F}(u_\lambda,\lambda)=0\), then \[\langle \mathcal{A}(u_\lambda),v\rangle = \lambda\langle\mathcal{G}(u_\lambda),v\rangle \qquad \forall v\in\mathcal{S}\setminus\{0\}.\] Hence \(\mathcal{R}(u_\lambda,v)=\lambda\) for all \(v\in\mathcal{S}\setminus\{0\}\), and so \(\lambda(u_\lambda)=\lambda\). Substituting \(u=u_\lambda\) into ?? gives ?? . ◻
Thus each pair \((u,v)\in\mathcal{U}^o\times(\mathcal{S}\setminus\{0\})\) with \(\Lambda(v)<+\infty\) gives the certified localization \[\lambda^*\in[\lambda(u),\Lambda(v)].\] The width of this interval is \[{\rm gap}(u,v):=\Lambda(v)-\lambda(u).\] If the minimax equality \(\lambda^*=\Lambda^*\) is not known, this width may also contain the duality gap \(\Lambda^*-\lambda^*\). Nevertheless, a small value of \({\rm gap}(u,v)\) still gives a computable a posteriori certificate for the critical parameter.
Remark 9. Estimate ?? is particularly useful in threshold problems, such as pull-in instability, voltage collapse, and tipping phenomena; see, for instance, [15], [17]–[21]. In such problems one wants to estimate how close a given admissible solution \((u_\lambda,\lambda)\) is to the loss of solvability.
For an operator \(\mathcal{A}\), set \[\mathcal{F}_{\mathcal{A}}(u,\lambda) := \mathcal{A}(u)-\lambda\mathcal{G}(u), \qquad \mathcal{R}_{\mathcal{A}}(u,v) := \frac{\langle\mathcal{A}(u),v\rangle}{\langle\mathcal{G}(u),v\rangle}.\]
Corollary 10 (One-sided perturbation bound). Let \(\mathcal{P}:\mathcal{U}^o\to W^*\), and assume that \[-\infty< \lambda^*_{\mathcal{A}+\mathcal{P}} := \sup_{u\in\mathcal{U}^o} \inf_{v\in\mathcal{S}\setminus\{0\}} \mathcal{R}_{\mathcal{A}+\mathcal{P}}(u,v) <+\infty .\] If \(u^*_{\mathcal{A}}\in\mathcal{U}^o\) satisfies \[\lambda^*_{\mathcal{A}} = \inf_{v\in\mathcal{S}\setminus\{0\}} \mathcal{R}_{\mathcal{A}}(u^*_{\mathcal{A}},v),\] then \[\label{eq:PertEST} \lambda^*_{\mathcal{A}+\mathcal{P}} \ge \lambda^*_{\mathcal{A}} + \inf_{v\in\mathcal{S}\setminus\{0\}} \frac{\langle \mathcal{P}(u^*_{\mathcal{A}}),v\rangle}{\langle\mathcal{G}(u^*_{\mathcal{A}}),v\rangle}.\qquad{(6)}\]
Proof. Since \[\mathcal{R}_{\mathcal{A}+\mathcal{P}}(u,v) = \mathcal{R}_{\mathcal{A}}(u,v) + \frac{\langle \mathcal{P}(u),v\rangle}{\langle\mathcal{G}(u),v\rangle},\] we test the outer supremum in the definition of \(\lambda^*_{\mathcal{A}+\mathcal{P}}\) at \(u=u^*_{\mathcal{A}}\). Using \(\inf(a+b)\ge\inf a+\inf b\), we obtain \[\lambda^*_{\mathcal{A}+\mathcal{P}} \ge \inf_{v\in\mathcal{S}\setminus\{0\}} \mathcal{R}_{\mathcal{A}+\mathcal{P}}(u^*_{\mathcal{A}},v) \ge \lambda^*_{\mathcal{A}} + \inf_{v\in\mathcal{S}\setminus\{0\}} \frac{\langle \mathcal{P}(u^*_{\mathcal{A}}),v\rangle}{\langle\mathcal{G}(u^*_{\mathcal{A}}),v\rangle}.\] This proves ?? . ◻
Proof of Theorem 4. We first record two elementary identities used below. Since \[\mathcal{R}(u,v) = \frac{\langle\mathcal{A}(u),v\rangle}{\langle\mathcal{G}(u),v\rangle}, \qquad \mathcal{F}(u,\lambda)=\mathcal{A}(u)-\lambda\mathcal{G}(u),\] the quotient rule gives, whenever \(\langle\mathcal{G}(u),v\rangle\ne0\), \[D_u\mathcal{R}(u,v)(\xi) = \frac{ \bigl\langle D_u\mathcal{F}(u,\mathcal{R}(u,v))\xi,v \bigr\rangle }{ \langle\mathcal{G}(u),v\rangle }, \qquad D_v\mathcal{R}(u,v)(\zeta) = \frac{ \bigl\langle \mathcal{F}(u,\mathcal{R}(u,v)),\zeta \bigr\rangle }{ \langle\mathcal{G}(u),v\rangle }.\] Thus, at a point where \(\mathcal{R}(u,v)=\lambda\), the condition \(D_u\mathcal{R}(u,v)=0\) is equivalent to \[\bigl\langle D_u\mathcal{F}(u,\lambda)\xi,v\bigr\rangle=0 \qquad \forall \xi,\] whereas \(D_v\mathcal{R}(u,v)=0\) is equivalent to \[\bigl\langle \mathcal{F}(u,\lambda),\zeta\bigr\rangle=0 \qquad \forall \zeta .\]
We first prove \((1^\circ)\). Fix \(r\) sufficiently large and set \[\lambda_r(u):= \inf_{v\in\mathcal{S}_r\setminus\{0\}}\mathcal{R}(u,v), \qquad u\in\mathcal{S}_r^o .\] Let \(\{\eta_1,\ldots,\eta_{N_r}\}\) be the basis associated with \(\mathcal{S}_r\) in \({\rm (c2)}\). Then, by \({\rm (CQ)}\), \(b_i(u):=\langle\mathcal{G}(u),\eta_i\rangle>0\). Hence, for \(v=\sum_i v^i\eta_i\in\mathcal{S}_r\setminus\{0\}\), \(v^i\ge0\), we have \[\mathcal{R}(u,v) = \sum_{i=1}^{N_r} \frac{v^i b_i(u)}{\sum_j v^j b_j(u)} \mathcal{R}(u,\eta_i),\] and therefore \[\lambda_r(u)=\min_{1\le i\le N_r}\mathcal{R}(u,\eta_i).\] By the \(C^1\)-part of \({\rm (CQ)}\), the functions \(u\mapsto\mathcal{R}(u,\eta_i)\) are \(C^1\) on \(\mathcal{S}_r^o\). Hence \(\lambda_r\) is continuous on \(\mathcal{S}_r^o\).
By \({\rm (h1)}\), \(\lambda_r\) attains its maximum. Indeed, any maximizing sequence eventually belongs to \(K_{r,\alpha_r}\), whose closure is compact and contained in \(\mathcal{S}_r^o\). Hence there exists \(u_r^*\in\mathcal{S}_r^o\) such that \[\lambda_r^* = \lambda_r(u_r^*) = \inf_{v\in\mathcal{S}_r\setminus\{0\}}\mathcal{R}(u_r^*,v).\]
The point \((u_r^*,\lambda_r^*)\) solves the finite-dimensional constrained problem \[\text{maximize }\lambda \quad\text{over }(u,\lambda)\in\mathcal{S}_r^o\times\mathbb{R}, \qquad \lambda-\mathcal{R}(u,\eta_i)\le0,\quad i=1,\ldots,N_r .\] Since \(u_r^*\) is an interior point of \(\mathcal{S}_r^o\), no boundary multiplier in the \(u\)-variable appears. By the \(C^1\)-part of \({\rm (CQ)}\) and the Fritz John multiplier rule, there exist \(\mu_0\ge0\) and \(\mu_i\ge0\), not all zero, such that \[\mu_0=\sum_{i=1}^{N_r}\mu_i,\qquad \sum_{i=1}^{N_r}\mu_iD_u\mathcal{R}(u_r^*,\eta_i)=0, \qquad \mu_i\bigl(\lambda_r^*-\mathcal{R}(u_r^*,\eta_i)\bigr)=0 .\] Then \(\sum_i\mu_i>0\). After normalization, assume \(\sum_i\mu_i=1\), and set \[b_i:=\langle\mathcal{G}(u_r^*),\eta_i\rangle, \qquad v_r^*:= \sum_{i=1}^{N_r}\frac{\mu_i}{b_i}\eta_i .\] Then \(v_r^*\in\mathcal{S}_r\setminus\{0\}\). By the complementarity relations, \[\langle\mathcal{A}(u_r^*),v_r^*\rangle = \sum_i\mu_i\mathcal{R}(u_r^*,\eta_i) = \lambda_r^*, \qquad \langle\mathcal{G}(u_r^*),v_r^*\rangle=1,\] and hence \[\mathcal{R}(u_r^*,v_r^*)=\lambda_r^*.\]
For every \(\xi\in W_r\), again using complementarity and the identity for \(D_u\mathcal{R}\) recorded above, \[0 = \sum_i\mu_iD_u\mathcal{R}(u_r^*,\eta_i)(\xi) = \sum_i \frac{\mu_i}{b_i} \bigl\langle D_u\mathcal{F}(u_r^*,\lambda_r^*)\xi,\eta_i \bigr\rangle .\] Therefore \[\bigl\langle D_u\mathcal{F}(u_r^*,\lambda_r^*)\xi,v_r^* \bigr\rangle=0 \qquad \forall \xi\in W_r .\] Since \(\mathcal{R}(u_r^*,v_r^*)=\lambda_r^*\), this is equivalent to \(D_u\mathcal{R}(u_r^*,v_r^*)(\xi)=0\) for all \(\xi\in W_r\). Hence, by \({\rm (h2)}\), \(v_r^*\in\mathcal{S}_r^o\).
Thus \(v_r^*\) is an interior minimizer of \(v\mapsto\mathcal{R}(u_r^*,v)\). Consequently, \[D_v\mathcal{R}(u_r^*,v_r^*)(\zeta)=0 \qquad \forall \zeta\in W_r .\] Since \(\mathcal{R}(u_r^*,v_r^*)=\lambda_r^*\), the identity for \(D_v\mathcal{R}\) gives \[\langle\mathcal{F}(u_r^*,\lambda_r^*),\zeta\rangle=0 \qquad \forall \zeta\in W_r .\]
Thus \((u_r^*,v_r^*)\) is a finite-dimensional saddle-point pair at the level \(\lambda_r^*\). Moreover, \[\lambda_r^* = \mathcal{R}(u_r^*,v_r^*) = \min_{v\in\mathcal{S}_r^o}\mathcal{R}(u_r^*,v) = \max_{u\in\mathcal{S}_r^o} \min_{v\in\mathcal{S}_r^o}\mathcal{R}(u,v).\] This proves \((1^\circ)\).
We prove \((2^\circ)\). Since the Galerkin scheme is admissible, \({\rm (CD)}\) holds. Suppose that, along a subsequence, \(\lambda_r^*\to\hat{\lambda}^*\). By \((1^\circ)\), \[\mathcal{R}(u_r^*,v_r^*)=\lambda_r^*\to\hat{\lambda}^*, \qquad D_u\mathcal{R}(u_r^*,v_r^*)=0\;\text{on }W_r, \qquad D_v\mathcal{R}(u_r^*,v_r^*)=0\;\text{on }W_r .\] Thus \((u_r^*,v_r^*)\) is an extended \((PS)_e\)-sequence at level \(\hat{\lambda}^*\). By the extended \((PS)_e\)-condition, after a further subsequence and a positive rescaling of \(v_r^*\), \[u_r^*\to u^* \quad\text{strongly in }W, \qquad v_r^*\rightharpoonup v^* \quad\text{weakly in }W,\] with \(u^*\in\mathcal{S}^o\) and \(v^*\in\mathcal{S}\setminus\{0\}\).
The rescaling of \(v_r^*\) preserves the identities involving \(D_u\mathcal{F}\), while the Galerkin equation is independent of \(v_r^*\). Let \(\zeta\in W\), and choose \(\zeta_r\in W_r\) such that \(\zeta_r\to\zeta\) strongly in \(W\). Since \[\langle\mathcal{F}(u_r^*,\lambda_r^*),\zeta_r\rangle=0,\] the sequential closedness of the residual gives \[\mathcal{F}(u^*,\hat{\lambda}^*)=0 \quad\text{in }W^* .\] By \({\rm (R)}\), we then have \(u^*\in\mathcal{U}^o\).
Let \(\xi\in W\), and choose \(\xi_r\in W_r\) such that \(\xi_r\to\xi\) strongly in \(W\). Since \[\bigl\langle D_u\mathcal{F}(u_r^*,\lambda_r^*)\xi_r,v_r^* \bigr\rangle=0,\] condition \({\rm (CD)}\) yields \[\bigl\langle D_u\mathcal{F}(u^*,\hat{\lambda}^*)\xi,v^* \bigr\rangle=0 \qquad \forall \xi\in W .\] This proves \((2^\circ)\). ◻
Proof of Theorem 5. Recall that \[\lambda^* = \sup_{u\in\mathcal{U}^o} \lambda(u), \qquad \lambda(u):= \inf_{v\in\mathcal{S}\setminus\{0\}}\mathcal{R}(u,v).\] We first prove the liminf estimate \[\label{eq:liminf-Th2} \liminf_{r\to\infty}\lambda_r^*\ge \lambda^* .\tag{7}\] Let \(\varepsilon\in(0,\delta_0)\). By the definition of \(\lambda^*\), there exists \(u_\varepsilon\in\mathcal{U}^o\) such that \[\lambda(u_\varepsilon)>\lambda^*-\varepsilon .\] By \({\rm (c3)}\), for all sufficiently large \(r\), \(\mathcal{I}_ru_\varepsilon\in\mathcal{S}_r^o\). Hence \[\lambda_r^* \ge \lambda_r(\mathcal{I}_ru_\varepsilon) = \inf_{v\in\mathcal{S}_r\setminus\{0\}} \mathcal{R}(\mathcal{I}_ru_\varepsilon,v).\] Taking the lower limit and using \({\rm (LC)}\), we obtain \[\liminf_{r\to\infty}\lambda_r^* \ge \lambda(u_\varepsilon) > \lambda^*-\varepsilon .\] Letting \(\varepsilon\downarrow0\), we get 7 .
By the assumed bounds on \(\lambda_r^*\), every subsequence of \((\lambda_r^*)\) has a further convergent subsequence. Let \[\lambda_{r_j}^*\to\bar\lambda, \qquad \bar\lambda\in[c,C].\] Then 7 gives \(\lambda^*\le\bar\lambda\).
By Theorem 4\((1^\circ)\), for every sufficiently large \(j\) there exist Galerkin minimax pairs \[(u_{r_j}^*,v_{r_j}^*) \in \mathcal{S}_{r_j}^o\times\mathcal{S}_{r_j}^o .\] Since \(\mathcal{R}\) satisfies the extended \((PS)_e\)-condition at every level in \([c,C]\), Theorem 4\((2^\circ)\) yields, after passing to a further subsequence and rescaling the second component, a pair \[\bar u\in\mathcal{U}^o, \qquad \bar v\in\mathcal{S}\setminus\{0\},\] such that \[u_{r_j}^*\to\bar u \quad\text{strongly in }W, \qquad v_{r_j}^*\rightharpoonup\bar v \quad\text{weakly in }W,\] and \[\mathcal{F}(\bar u,\bar\lambda)=0 \quad\text{in }W^*, \qquad \bigl\langle D_u\mathcal{F}(\bar u,\bar\lambda)\xi,\bar v \bigr\rangle=0 \quad \forall \xi\in W .\] Since \(\bar u\in\mathcal{U}^o\subset\mathcal{S}^o\), condition \({\rm (D)}\) applies. Therefore \[\mathcal{R}(\bar u,v)=\bar\lambda \qquad \forall v\in\mathcal{S}\setminus\{0\},\] and hence \[\bar\lambda = \inf_{v\in\mathcal{S}\setminus\{0\}}\mathcal{R}(\bar u,v) \le \sup_{u\in\mathcal{U}^o} \inf_{v\in\mathcal{S}\setminus\{0\}}\mathcal{R}(u,v) = \lambda^* .\] Together with \(\lambda^*\le\bar\lambda\), this gives \(\bar\lambda=\lambda^*\).
Thus every subsequence of \((\lambda_r^*)\) has a further subsequence converging to \(\lambda^*\). Consequently, \(\lambda_r^*\to\lambda^*\). In particular, \(\lambda^*\in[c,C]\), and hence \(\lambda^*>0\).
Applying the preceding compactness argument to the convergent sequence \(\lambda_r^*\to\lambda^*\), we obtain, after passing to a subsequence and rescaling \(v_r^*\), a pair \[(u^*,v^*)\in\mathcal{U}^o\times(\mathcal{S}\setminus\{0\})\] such that \[u_r^*\to u^* \quad\text{strongly in }W, \qquad v_r^*\rightharpoonup v^* \quad\text{weakly in }W,\] and \[\mathcal{F}(u^*,\lambda^*)=0 \quad\text{in }W^*, \qquad \bigl\langle D_u\mathcal{F}(u^*,\lambda^*)\xi,v^* \bigr\rangle=0 \quad \forall \xi\in W .\] Again, since \(u^*\in\mathcal{U}^o\subset\mathcal{S}^o\), condition \({\rm (D)}\) gives \[\mathcal{R}(u^*,v)=\lambda^* \qquad \forall v\in\mathcal{S}\setminus\{0\}.\] In particular, \(\mathcal{R}(u^*,v^*)=\lambda^*\), and therefore \[\lambda^* = \mathcal{R}(u^*,v^*) = \inf_{v\in\mathcal{S}\setminus\{0\}}\mathcal{R}(u^*,v) = \sup_{u\in\mathcal{U}^o} \inf_{v\in\mathcal{S}\setminus\{0\}} \mathcal{R}(u,v).\] Thus the minimax bifurcation formula holds, and \((u^*,v^*)\) is a saddle-point solution pair at the level \(\lambda^*\).
It remains to identify this value as the maximal one-sided threshold. Indeed, if \(u\in\mathcal{U}^o\) is a solution of 3 at some parameter \(\lambda\), then condition \({\rm (D)}\) gives \[\mathcal{R}(u,v)=\lambda \qquad \forall v\in\mathcal{S}\setminus\{0\}.\] Hence \(\lambda=\lambda(u)\le\lambda^*\). Therefore 3 has no solution in \(\mathcal{U}^o\) for any \(\lambda>\lambda^*\).
Finally, every one-sided saddle-node point in \(\mathcal{U}^o\) is, by definition, a solution of 3 . Hence its parameter cannot exceed \(\lambda^*\). Since \((u^*,\lambda^*)\) is a singular solution and no admissible solutions exist above \(\lambda^*\), it is a maximal one-sided saddle-node point in \(\mathcal{U}^o\).
The convergence of the Galerkin minimax levels and, along a subsequence, of the corresponding minimax pairs has already been proved. ◻
We now apply the abstract minimax principle to non-variational elliptic systems. Let \(\Omega\subset\mathbb{R}^d\), \(d\ge1\), be a bounded domain, sufficiently regular for the Hopf boundary principle to apply. We also assume that \(\Omega\) admits an exhaustion from inside by polyhedral domains \(\Omega_r\subset\Omega\), \(r=1,2,\ldots\), as in Appendix 8. The associated conforming triangulations are chosen so that the discrete maximum principle 17 holds. The index \(r\) includes both the choice of the subdomain and the mesh refinement.
Let \(m\ge1\). We consider the system \[\label{f} \left\{ \begin{align} \mathcal{L} u^k &=\lambda (u^k)^{q-1}+f_k(x,u), && x\in\Omega,\quad k=1,\ldots,m,\\ u^k&\ge0, && x\in\Omega,\quad k=1,\ldots,m,\\ u^k&=0, && x\in\partial\Omega,\quad k=1,\ldots,m, \end{align} \right.\tag{8}\] where \(u=(u^1,\ldots,u^m)\) and \(1<q<2\). For simplicity, the same scalar elliptic operator acts in each component: \[\label{eq:mathL} \mathcal{L} w = -\frac{\partial}{\partial x_\alpha} \left( \sigma_{\alpha\beta}(x) \frac{\partial w}{\partial x_\beta} \right) +c(x)w ,\tag{9}\] where summation over repeated spatial indices is understood. We assume that \(\sigma_{\alpha\beta},c\in C(\overline{\Omega})\), that \((\sigma_{\alpha\beta})\) is symmetric and uniformly elliptic, and that the bilinear form \[\label{eq:awz} a(w,z):= \int_\Omega \left( \sigma_{\alpha\beta}(x) \frac{\partial w}{\partial x_\beta} \frac{\partial z}{\partial x_\alpha} +c(x)wz \right)\,dx\tag{10}\] is coercive on \(H_0^1(\Omega)\). Set \[\mathbb{W}:=(H_0^1(\Omega))^m, \qquad \mathbb{L}^p:=(L^p(\Omega))^m,\] and \[a_m(u,v):=\sum_{k=1}^m a(u^k,v^k), \qquad \langle u,v\rangle_m:=\sum_{k=1}^m\int_\Omega u^k v^k\,dx .\]
We write \[\mathcal{L} u:=(\mathcal{L} u^1,\ldots,\mathcal{L} u^m)^T, \qquad f(u):=(f_1(\cdot,u),\ldots,f_m(\cdot,u))^T,\] and \[g(u):=\bigl((u^1)^{q-1},\ldots,(u^m)^{q-1}\bigr)^T .\] Then 8 has the abstract form \[\mathcal{F}(u,\lambda):=\mathcal{L} u-f(u)-\lambda g(u)=0.\]
We impose the following assumptions. For each \(i=1,\ldots,m\), \(f_i(\cdot,u)\in C(\overline{\Omega})\), \(f_i(x,\cdot)\in C^1(\mathbb{R}_+^m)\), and \(f_i(x,0)=0\). Moreover, there exist \(2<\gamma_1\le\gamma_2<2^*\) and \(C>0\) such that, for \(i,j=1,\ldots,m\), \[\begin{align} 0\le f_i(x,u) &\le C\bigl(|u|^{\gamma_1-1}+|u|^{\gamma_2-1}\bigr), \tag{11} \\ 0\le f_{i,u^j}(x,u) &\le C\bigl(|u|^{\gamma_1-2}+|u|^{\gamma_2-2}\bigr), \tag{12} \end{align}\] and, for \(i\ne j\), \[f_{i,u^j}(x,u)>0 \qquad \text{for a.e. }x\in\Omega,\quad u\in(0,+\infty)^m .\]
We also assume the following large-growth condition.
There exist \(R>0\) and \(\delta_0>0\) such that, denoting by \(\lambda_1\) the principal Dirichlet eigenvalue of \(\mathcal{L}\), \[\sum_{k=1}^m f_k(x,u) \ge (\lambda_1+\delta_0)\sum_{k=1}^m u^k\] for a.e. \(x\in\Omega\) and all \(u\in\mathbb{R}_+^m\) with \(|u|>R\).
Finally, we require the radial monotonicity condition \[\label{eq:radial-monotonicity-f} \bigl\langle f_u(u)u-f(u),v\bigr\rangle_m\ge0 \qquad u\in\mathbb{U}^o,\quad v\in\mathbb{S}\setminus\{0\},\tag{13}\] and, for some \(\theta>1\), the Picone-type superlinearity condition \[\label{eq:picone-superlinearity-f} Q_f(x,u)\ge \theta\sum_{i=1}^m f_i(x,u)u^i \qquad \text{for a.e. }x\in\Omega,\quad u\in\mathbb{R}_+^m,\tag{14}\] where \[Q_f(x,u):= \sum_{i=1}^m f_{i,u^i}(x,u)(u^i)^2 + 2\sum_{1\le i<j\le m} \sqrt{f_{i,u^j}(x,u)f_{j,u^i}(x,u)}\,u^iu^j .\]
We use the scalar cone \[S^o := \left\{ w\in H_0^1(\Omega)\cap C(\overline{\Omega}): w>0 \;\text{in }\Omega \right\}, \qquad S:=\overline{S^o}^{\,C(\overline{\Omega})}\cap H_0^1(\Omega),\] and its componentwise version \[\mathbb{S}^o:=(S^o)^m, \qquad \mathbb{S}:=\overline{\mathbb{S}^o}^{\,\mathbb{W}}.\] The regular cone is \(\mathbb{U}^o:=(U^o)^m\), where \[U^o := \left\{ w\in S^o: \mathcal{L}w\in L^\infty_{\rm loc}(\Omega), \quad \mathcal{L}w\ge0 \text{ in }\mathcal{D}'(\Omega) \right\}.\]
By the Hopf-type lower bound, every \(w\in U^o\) satisfies \[w\ge c_w d_\Omega \quad\text{in }\Omega, \qquad d_\Omega(x):=\operatorname{dist}(x,\partial\Omega),\] with some \(c_w>0\). Hence Lemma 22 implies that \(u\mapsto g(u)\) is differentiable on \(\mathbb{U}^o\), and \(D_u\mathcal{F}(u,\lambda)\) is well defined for every \(u\in\mathbb{U}^o\).
We now verify the abstract regularity condition \({\rm (R)}\) at nonnegative levels, in the positive class considered here. Let \(u\in\mathbb{S}^o\) be a weak solution of 8 with \(\lambda\ge0\). Then, for each \(k=1,\ldots,m\), \[\mathcal{L} u^k = \lambda (u^k)^{q-1}+f_k(x,u) \in L^\infty_{\rm loc}(\Omega), \qquad \mathcal{L} u^k\ge0 \quad\text{in }\mathcal{D}'(\Omega).\] The strong maximum principle and the Hopf boundary principle therefore give \(u^k\in U^o\) for every \(k=1,\ldots,m\). Thus \(u\in\mathbb{U}^o\).
The extended Rayleigh quotient associated with 8 is \[\mathcal{R}(u,v):= \frac{ a_m(u,v)-\langle f(u),v\rangle_m }{ \langle g(u),v\rangle_m }, \qquad u\in\mathbb{S}^o,\quad v\in\mathbb{S}\setminus\{0\}.\] Its denominator is positive on this set. The corresponding infinite-dimensional minimax level is \[\lambda^* := \sup_{u\in\mathbb{U}^o} \inf_{v\in\mathbb{S}\setminus\{0\}} \mathcal{R}(u,v).\] We use the finite-dimensional Galerkin cones constructed in Appendix 8. Thus \(W_r\subset H_0^1(\Omega_r)\) is extended by zero to \(\Omega\), so that \(W_r\subset H_0^1(\Omega)\), and \(S_r^o\) is the corresponding positive nodal cone. We use the product spaces and cones \[\mathbb{W}_r:=(W_r)^m, \qquad \mathbb{S}_r^o:=(S_r^o)^m, \qquad \mathbb{S}_r:=(S_r)^m .\] The componentwise interpolation operator is denoted again by \(\mathcal{I}_r\). The finite-dimensional minimax levels are \[\lambda_r^* := \sup_{u\in\mathbb{S}_r^o} \inf_{v\in\mathbb{S}_r\setminus\{0\}} \mathcal{R}(u,v).\]
Lemma 11 (Uniform bounds for the minimax levels). Assume the hypotheses stated above, including \({\rm (LG)}\), and suppose that the Galerkin cones and the discrete maximum-principle structure are those described in Appendix 8. Then \[\lambda^*<+\infty .\] Moreover, there exist constants \(c,C>0\) and \(r_0\ge1\), independent of \(r\), such that \[c\le \lambda_r^*\le C, \qquad r\ge r_0 .\]
Proof. Let \((\lambda_1,\phi_1)\) be the principal Dirichlet eigenpair of \(\mathcal{L}\), with \(\phi_1>0\) in \(\Omega\). Since \(\mathcal{L}\) is symmetric, \[a(w,\phi_1)=\lambda_1\int_\Omega w\phi_1\,dx \qquad \forall w\in H_0^1(\Omega).\] Set \(\boldsymbol{\phi}_1:=(\phi_1,\ldots,\phi_1)\). Testing the inner infimum by \(\boldsymbol{\phi}_1\), we get \[\lambda^* \le \sup_{u\in\mathbb{U}^o} \frac{ \displaystyle \int_\Omega \left( \lambda_1\sum_{k=1}^m u^k - \sum_{k=1}^m f_k(x,u) \right)\phi_1\,dx }{ \displaystyle \int_\Omega \sum_{k=1}^m (u^k)^{q-1}\phi_1\,dx }.\] For \(\mu\ge0\), define \[H_\mu(x,u):= \frac{ \mu\sum_{k=1}^m u^k-\sum_{k=1}^m f_k(x,u) }{ \sum_{k=1}^m (u^k)^{q-1} }, \qquad u\in\mathbb{R}_+^m\setminus\{0\}.\] By the growth assumptions on \(f\), \(1<q<2\), and \({\rm (LG)}\), \[\sup_{x\in\Omega,\;u\in\mathbb{R}_+^m\setminus\{0\}} H_{\lambda_1}(x,u)<+\infty .\] Hence \(\lambda^*<+\infty\).
The discrete upper bound is identical. Let \((\lambda_{1,r},\phi_{1,r})\) be the discrete principal eigenpair. By admissibility, \[\lambda_{1,r}\to\lambda_1, \qquad \phi_{1,r}\in S_r^o .\] Thus, for all sufficiently large \(r\), \(\lambda_{1,r}\le\lambda_1+\delta_0/2\). The same pointwise bound for \(H_\mu\), uniformly for \(\mu\in[0,\lambda_1+\delta_0/2]\), and the test \(\boldsymbol{\phi}_{1,r}:=(\phi_{1,r},\ldots,\phi_{1,r})\in\mathbb{S}_r^o\) give \[\lambda_r^*\le C .\]
It remains to prove the positive lower bound. Let \(\omega_r\in W_r\) be the discrete torsion function, \[a(\omega_r,\psi_i)=\int_\Omega\psi_i\,dx, \qquad i=1,\ldots,N_r .\] By the discrete maximum principle and the admissibility assumptions, \[\omega_r\in S_r^o, \qquad \|\omega_r\|_{L^\infty(\Omega)}\le C ,\] with \(C\) independent of \(r\). Set \(\boldsymbol{\omega}_r:=(\omega_r,\ldots,\omega_r)\). For \(0<t\le1\) and \(v\in\mathbb{S}_r\setminus\{0\}\), the definition of \(\omega_r\) gives \[\frac{a_m(t\boldsymbol{\omega}_r,v)}{\langle g(t\boldsymbol{\omega}_r),v\rangle_m} \ge c_1t^{2-q}.\] On the other hand, using \(1<q<2<\gamma_1\le\gamma_2\) and \(\|\omega_r\|_{L^\infty}\le C\), \[\frac{\langle f(t\boldsymbol{\omega}_r),v\rangle_m}{\langle g(t\boldsymbol{\omega}_r),v\rangle_m} \le c_2t^{\gamma_1-q}+c_3t^{\gamma_2-q}.\] Therefore \[\mathcal{R}(t\boldsymbol{\omega}_r,v) \ge c_1t^{2-q} - c_2t^{\gamma_1-q} - c_3t^{\gamma_2-q}.\] Since \(\gamma_1>2\), we fix \(t_0\in(0,1]\), independent of \(r\), such that the right-hand side is bounded below by some \(c>0\). Hence \[\lambda_r^* \ge \inf_{v\in\mathbb{S}_r\setminus\{0\}} \mathcal{R}(t_0\boldsymbol{\omega}_r,v) \ge c\] for all sufficiently large \(r\). The proof is complete. ◻
Lemma 12. Under the assumptions stated above, suppose in addition that the discrete comparison estimate of Lemma 19 holds for the Galerkin operator \(A_r\). Then \(\mathcal{R}\) satisfies the extended \((PS)_e\)-condition at every level \(\lambda>0\).
Proof. Let \(r_n\to\infty\), and let \((u_n,v_n)\in\mathbb{S}_{r_n}^o\times\mathbb{S}_{r_n}^o\) be an extended \((PS)_e\)-sequence at a level \(\lambda>0\). Set \[\lambda_n:=\mathcal{R}(u_n,v_n)\to\lambda .\] By the homogeneity of \(\mathcal{R}\) with respect to the second variable, we normalize \(a_m(v_n,v_n)=1\). The Galerkin criticality relations, equivalent to \(D_u\mathcal{R}(u_n,v_n)=D_v\mathcal{R}(u_n,v_n)=0\) on \(\mathbb{W}_{r_n}\), are \[\langle\mathcal{F}(u_n,\lambda_n),\zeta\rangle=0 \quad \forall \zeta\in\mathbb{W}_{r_n}, \qquad \bigl\langle D_u\mathcal{F}(u_n,\lambda_n)\xi,v_n\bigr\rangle=0 \quad \forall \xi\in\mathbb{W}_{r_n}.\] Testing the first identity by \(u_n\), we get \[\label{eq:energy-ps-g} a_m(u_n,u_n) = \lambda_n\langle g(u_n),u_n\rangle_m + \langle f(u_n),u_n\rangle_m .\tag{15}\]
We first prove that \((u_n)\) is bounded in \(\mathbb{W}\). Let \(\theta_n\in\mathbb{W}_{r_n}\) be the nodal function defined componentwise by \[(\theta_n)_i^k=\frac{(u_{n,i}^k)^2}{v_{n,i}^k}.\] Since \(u_n,v_n\in\mathbb{S}_{r_n}^o\), this function is well defined. Taking \(\xi=\theta_n\) in the adjoint Galerkin identity, using the componentwise discrete Picone inequality, and applying the arithmetic–geometric mean inequality to the cooperative off-diagonal terms, we obtain \[a_m(u_n,u_n) \ge (q-1)\lambda_n\langle g(u_n),u_n\rangle_m + \int_\Omega Q_f(x,u_n)\,dx .\] By 14 , for some \(\eta>1\), \[a_m(u_n,u_n) \ge (q-1)\lambda_n\langle g(u_n),u_n\rangle_m + \eta\langle f(u_n),u_n\rangle_m .\] Combining this estimate with 15 , we find \[(\eta-1)a_m(u_n,u_n) \le (\eta-q+1)\lambda_n\langle g(u_n),u_n\rangle_m .\] Since \[\langle g(u_n),u_n\rangle_m = \|u_n\|_{\mathbb{L}^q}^q \le Ca_m(u_n,u_n)^{q/2},\] and \(q<2\), the sequence \((u_n)\) is bounded in \(\mathbb{W}\).
By the normalization of \(v_n\) and the coercivity of \(a_m\), the sequence \((v_n)\) is also bounded in \(\mathbb{W}\). Hence, after passing to a subsequence, \[u_n\rightharpoonup u^*, \qquad v_n\rightharpoonup v^* \quad\text{weakly in }\mathbb{W},\] and \[u_n\to u^*, \qquad v_n\to v^* \quad\text{strongly in }\mathbb{L}^\beta,\quad 1\le\beta<2^*,\] and a.e. in \(\Omega\). Since \(\mathbb{S}\) is weakly closed and \(\mathbb{S}_{r_n}\subset\mathbb{S}\), we have \(u^*,v^*\in\mathbb{S}\).
We now pass to the limit in the Galerkin equation. Let \(\zeta\in\mathbb{W}\). Choose \(\zeta_n\in\mathbb{W}_{r_n}\) such that \[\zeta_n\to\zeta \qquad\text{strongly in }\mathbb{W} .\] Then \[\langle\mathcal{F}(u_n,\lambda_n),\zeta_n\rangle=0.\] Using the weak convergence of \(u_n\), the strong convergence in subcritical Lebesgue spaces, and the growth assumptions on \(f\), we obtain \[\langle\mathcal{F}(u^*,\lambda),\zeta\rangle=0 \qquad \forall \zeta\in\mathbb{W} .\]
We next show that \(u^*\in\mathbb{S}^o\). Since \(\lambda_n\to\lambda>0\), for all large \(n\), \[\mathcal{L} u_n^k = \lambda_n(u_n^k)^{q-1}+f_k(x,u_n) \ge \frac{\lambda}{2}(u_n^k)^{q-1}\] in the discrete weak sense, for \(k=1,\ldots,m\). By Lemma 19, applied componentwise, \[u_n^k\ge w_n^k,\] where \(w_n^k\in S_{r_n}^o\) is the Galerkin solution of the scalar sublinear problem \[\mathcal{L} w_n^k=\frac{\lambda}{2}(w_n^k)^{q-1}.\] The standard convergence of Galerkin solutions for this scalar sublinear problem gives \[w_n^k\to w \quad\text{strongly in }H_0^1(\Omega) \quad\text{and a.e. in }\Omega,\] where \(w\) is the positive weak solution of \[\mathcal{L} w=\frac{\lambda}{2}w^{q-1}, \qquad w\in H_0^1(\Omega), \qquad w>0 \quad\text{in }\Omega .\] Passing to the limit in \(u_n^k\ge w_n^k\), we obtain \[u^{*,k}\ge w>0 \quad\text{a.e. in }\Omega, \qquad k=1,\ldots,m .\] Since \(u^*\) solves the system and the right-hand side has subcritical growth, standard bootstrap/Moser estimates and boundary Hölder regularity imply \(u^*\in(C(\overline{\Omega}))^m\). Hence \(u^{*,k}>0\) in \(\Omega\), \(k=1,\ldots,m\), and therefore \(u^*\in\mathbb{S}^o\).
It remains to strengthen the convergence of \(u_n\) and to prove that \(v^*\ne0\). Testing the limit equation by \(u^*\), we get \[a_m(u^*,u^*) = \lambda\langle g(u^*),u^*\rangle_m + \langle f(u^*),u^*\rangle_m .\] Comparing this identity with 15 , and using the compact convergence of the nonlinear terms, we obtain \[a_m(u_n,u_n)\to a_m(u^*,u^*) .\] Together with the weak convergence \(u_n\rightharpoonup u^*\), the coercivity of \(a_m\) gives \[u_n\to u^* \qquad\text{strongly in }\mathbb{W} .\]
Finally, taking \(\xi=v_n\) in the adjoint Galerkin identity and using the normalization \(a_m(v_n,v_n)=1\), we obtain \[1 = (q-1)\lambda_n \left\langle u_n^{q-2}v_n,v_n\right\rangle_m + \left\langle f_u(u_n)v_n,v_n\right\rangle_m .\] On the other hand, testing the Galerkin equation by the nodal function with values \[\frac{(v_{n,i}^k)^2}{u_{n,i}^k}\] and using the componentwise discrete Picone inequality yields \[1 \ge \lambda_n \left\langle u_n^{q-2}v_n,v_n\right\rangle_m + \left\langle f(u_n),\frac{v_n^2}{u_n}\right\rangle_m .\] Choose \(\chi\in(q-1,1)\). Multiplying the last inequality by \(\chi\) and subtracting it from the preceding identity gives \[1-\chi \le \left\langle f_u(u_n)v_n,v_n\right\rangle_m ,\] because the remaining terms are nonpositive: \[(q-1-\chi)\lambda_n \left\langle u_n^{q-2}v_n,v_n\right\rangle_m\le0, \qquad -\chi \left\langle f(u_n),\frac{v_n^2}{u_n}\right\rangle_m\le0 .\] If \(v^*=0\), then by compactness \[v_n\to0 \quad\text{strongly in }\mathbb{L}^\beta,\qquad 1\le\beta<2^* .\] Since \((u_n)\) is bounded in \(\mathbb{W}\) and \(f_u\) has subcritical growth, \(\left\langle f_u(u_n)v_n,v_n\right\rangle_m\to0\), contradicting the previous estimate. Hence \(v^*\ne0\).
Consequently, after passing to a subsequence, \[u_n\to u^* \quad\text{strongly in }\mathbb{W}, \qquad v_n\rightharpoonup v^* \quad\text{weakly in }\mathbb{W},\] with \(u^*\in\mathbb{S}^o\), \(v^*\in\mathbb{S}\setminus\{0\}\). This is precisely the extended \((PS)_e\)-condition at the level \(\lambda>0\). ◻
Lemma 13. Under the assumptions stated above, the extended Rayleigh quotient \(\mathcal{R}\) satisfies \({\rm (D)}\), the Galerkin scheme is admissible, and condition \({\rm (H)}\) holds for all sufficiently large \(r\).
Proof. The denominator condition \({\rm (D)}\) follows from the componentwise positivity of \(g(u)\) for \(u\in\mathbb{S}^o\) and from \(v\ge0\), \(v\ne0\), for \(v\in\mathbb{S}\setminus\{0\}\).
We next verify the Galerkin assumptions. Condition \({\rm (C)}\) is precisely the componentwise version of the nodal cone construction in Appendix 8. The quotient-compatibility condition \({\rm (CQ)}\) follows from the same denominator positivity on \(\mathbb{S}_r^o\times(\mathbb{S}_r\setminus\{0\})\), and from the finite-dimensional \(C^1\)-regularity of the restrictions of \(\mathcal{R}\) on \(\mathbb{S}_r^o\times\mathbb{S}_r^o\). The linearized closedness condition \({\rm (CD)}\) is verified in Appendix 11; the only singular term is the derivative of the concave part, and its passage to the limit follows from the discrete barrier estimate established for the Galerkin \((PS)_e\)-sequences.
We verify \({\rm (h1)}\). Fix \(r\) sufficiently large and choose \(0<\alpha<\lambda_r^*\). For \(u\in\mathbb{S}_r^o\), set \[\lambda_r(u):= \inf_{v\in\mathbb{S}_r\setminus\{0\}}\mathcal{R}(u,v).\] Since, for fixed \(u\), the quotient is linear-fractional in \(v\) with positive denominator, and since \(\mathbb{S}_r\) is generated by the componentwise nodal basis, we have \[\lambda_r(u) = \min_{1\le i\le N_r,\;1\le k\le m} \mathcal{R}(u,\bar\psi_i^k).\] Hence \(\lambda_r\) is continuous on \(\mathbb{S}_r^o\). Let \[K_{r,\alpha}:= \{u\in\mathbb{S}_r^o:\lambda_r(u)\ge\alpha\}.\] We show that \(K_{r,\alpha}\Subset\mathbb{S}_r^o\). The estimates below are finite-dimensional and use the equivalence of norms in \(\mathbb{W}_r\).
First, \(K_{r,\alpha}\) stays away from the origin. Indeed, if \(u_n\in K_{r,\alpha}\) and \(u_n\to0\) in \(\mathbb{W}_r\), then, choosing a nodal generator corresponding to a largest nodal coefficient of \(u_n\), one obtains \[\lambda_r(u_n)\le C\|u_n\|_{\mathbb{W}_r}^{\,2-q}+o(1)\to0,\] because \(1<q<2\), contradicting \(\lambda_r(u_n)\ge\alpha\).
Next, \(K_{r,\alpha}\) is bounded. Suppose, to the contrary, that \(\|u_n\|_{\mathbb{W}_r}\to\infty\). Set \[\rho_n:=\|u_n\|_{\mathbb{W}_r}, \qquad z_n:=\frac{u_n}{\rho_n}.\] Passing to a subsequence, \(z_n\to z_0\in\mathbb{S}_r\) with \(\|z_0\|_{\mathbb{W}_r}=1\). Testing the inner infimum by the componentwise discrete principal eigenfunction and using the large-growth condition \({\rm (LG)}\), we get \[\limsup_{n\to\infty}\lambda_r(u_n)\le0,\] again contradicting \(u_n\in K_{r,\alpha}\).
It remains to exclude convergence to the boundary of \(\mathbb{S}_r^o\). Let \(u_n\in K_{r,\alpha}\) and suppose that \(u_n\to u_0\in\partial\mathbb{S}_r^o\). The case \(u_0=0\) has already been excluded. Hence at least one component of \(u_0\) is nonzero.
If a nonzero component \(u_0^k\) has a zero nodal coefficient, then, by the irreducible \(M\)-matrix sign structure, there is a nodal generator \(\bar\psi_i^k\) such that \(a(u_0^k,\psi_i)<0\). Consequently, \[\limsup_{n\to\infty} \mathcal{R}(u_n,\bar\psi_i^k)<0,\] and therefore \(\limsup_{n\to\infty}\lambda_r(u_n)\le0\), which contradicts \(\lambda_r(u_n)\ge\alpha\).
It remains only to consider the case where an entire component \(u_0^\ell\equiv0\). Since \(u_0\ne0\), some other component is nonzero. Testing the inner infimum in the \(\ell\)-th component by a nodal generator corresponding to a largest nodal coefficient of \(u_n^\ell\), and using \(f_\ell\ge0\), gives \[\limsup_{n\to\infty}\lambda_r(u_n)\le0.\] This is again impossible. Hence \(K_{r,\alpha}\Subset\mathbb{S}_r^o\), and \({\rm (h1)}\) follows.
We now verify \({\rm (h2)}\). Let \(u\in\mathbb{S}_r^o\), \(v\in\mathbb{S}_r\setminus\{0\}\), and assume that \(\mathcal{R}(u,v)=\lambda_r^*<+\infty\) and \[D_u\mathcal{R}(u,v)(\xi)=0 \qquad \forall \xi\in\mathbb{W}_r .\] Then, by Lemma 11, \(\lambda_r^*\ge0\) for all sufficiently large \(r\), and hence \[a_m(\xi,v) = \langle f_u(u)\xi,v\rangle_m + \lambda_r^*(q-1)\langle u^{q-2}\xi,v\rangle_m \qquad \forall \xi\in\mathbb{W}_r .\] In particular, \(a_m(\xi,v)\ge0\) for all \(\xi\in\mathbb{S}_r\). Testing this inequality on the componentwise nodal generators gives \[A_r^T\bar v^k\ge0, \qquad k=1,\ldots,m,\] where \(\bar v^k\) is the vector of nodal coefficients of \(v^k\). Since \(A_r^T\) is again an irreducible nonsingular \(M\)-matrix, each component \(v^k\) is either identically zero or belongs to \(S_r^o\).
We show that zero components are impossible. Suppose that \(v^\ell\equiv0\). Since \(v\ne0\), there exists \(k_0\ne\ell\) such that \(v^{k_0}\in S_r^o\). Taking \(\xi\) with only the \(\ell\)-th component nonzero in the stationarity identity, we obtain \[0 = \sum_{i=1}^m \int_\Omega f_{i,u^\ell}(x,u)\,\xi^\ell v^i\,dx .\] All terms are nonnegative. However, by strict cooperativity, the term with \(i=k_0\) is strictly positive for a suitable \(\xi^\ell\in S_r\). This contradiction shows that no component of \(v\) can vanish. Therefore \(v^k\in S_r^o\) for every \(k=1,\ldots,m\), and hence \(v\in\mathbb{S}_r^o\). This proves \({\rm (h2)}\), and therefore \({\rm (H)}\). ◻
Theorem 14. Assume that the system 8 satisfies the assumptions stated above and that the Galerkin scheme is admissible in the sense of Appendix 8. Let \(\hat{\lambda}^*\) be a limit point of the sequence \((\lambda_r^*)\). Then the following assertions hold.
For all sufficiently large \(r\), the Galerkin problem admits a finite-dimensional saddle-point pair \((u_r^*,v_r^*)\in\mathbb{S}_r^o\times\mathbb{S}_r^o\) at the level \(\lambda_r^*\). Moreover, \[\lambda_r^* = \mathcal{R}(u_r^*,v_r^*) = \min_{v\in\mathbb{S}_r^o}\mathcal{R}(u_r^*,v) = \max_{u\in\mathbb{S}_r^o} \min_{v\in\mathbb{S}_r^o}\mathcal{R}(u,v).\]
Along a subsequence realizing \(\lambda_r^*\to\hat{\lambda}^*\), not relabelled, and after a positive rescaling of \(v_r^*\), there exist \(u^*\in\mathbb{U}^o\), \(v^*\in\mathbb{S}\setminus\{0\}\) such that \[u_r^*\to u^* \quad\text{strongly in }\mathbb{W}, \qquad v_r^*\rightharpoonup v^* \quad\text{weakly in }\mathbb{W} .\] Moreover, \(0<\hat{\lambda}^*\le\lambda^*\), and \[\mathcal{F}(u^*,\hat{\lambda}^*)=0 \quad\text{in }\mathbb{W}^*, \qquad \bigl\langle D_u\mathcal{F}(u^*,\hat{\lambda}^*)\xi,v^* \bigr\rangle=0 \quad \forall \xi\in\mathbb{W} .\] Thus \((u^*,\hat{\lambda}^*)\) is a weak singular solution of 8 .
Proof. By Lemma 13, the quotient \(\mathcal{R}\) satisfies \({\rm (D)}\), the Galerkin scheme is admissible, and condition \({\rm (H)}\) holds for all sufficiently large \(r\). Moreover, Lemma 11 gives \[\lambda^*<+\infty, \qquad 0<c_0\le\lambda_r^*\le C_0\] for all sufficiently large \(r\). Hence every limit point \(\hat{\lambda}^*\) of \((\lambda_r^*)\) is finite and positive. Applying Theorem 4\((1^\circ)\), we obtain assertion \((1^\circ)\).
Let a subsequence, not relabelled, be chosen so that \[\lambda_r^*\to\hat{\lambda}^* .\] Since \(\hat{\lambda}^*>0\), Lemma 12 applies at the level \(\hat{\lambda}^*\). Therefore Theorem 4\((2^\circ)\) yields, after passing to a further subsequence and after a positive rescaling of \(v_r^*\), \[u_r^*\to u^* \quad\text{strongly in }\mathbb{W}, \qquad v_r^*\rightharpoonup v^* \quad\text{weakly in }\mathbb{W},\] with \(u^*\in\mathbb{S}^o\), \(v^*\in\mathbb{S}\setminus\{0\}\), and \[\mathcal{F}(u^*,\hat{\lambda}^*)=0 \quad\text{in }\mathbb{W}^*, \qquad \bigl\langle D_u\mathcal{F}(u^*,\hat{\lambda}^*)\xi,v^* \bigr\rangle=0 \quad \forall \xi\in\mathbb{W} .\] By the regularity condition \({\rm (R)}\), this weak solution belongs to the regular cone, that is, \(u^*\in\mathbb{U}^o\).
Since \(u^*\in\mathbb{U}^o\subset\mathbb{S}^o\), condition \({\rm (D)}\) applies. From \[\mathcal{F}(u^*,\hat{\lambda}^*)=0\] we therefore obtain \[\mathcal{R}(u^*,v)=\hat{\lambda}^* \qquad \forall v\in\mathbb{S}\setminus\{0\}.\] Hence \[\hat{\lambda}^* = \inf_{v\in\mathbb{S}\setminus\{0\}}\mathcal{R}(u^*,v) \le \sup_{u\in\mathbb{U}^o} \inf_{v\in\mathbb{S}\setminus\{0\}}\mathcal{R}(u,v) = \lambda^* .\] Thus \[0<\hat{\lambda}^*\le\lambda^* .\] The remaining assertions in \((2^\circ)\) have already been obtained above, and the proof is complete. ◻
Remark 15 (On the role of symmetry). The symmetry of the principal part of \(\mathcal{L}\) is used only in the compactness argument, through the Picone-type inequalities in Appendix 9. The abstract minimax construction and the Galerkin saddle-point argument are not intrinsically symmetric. Nonsymmetric operators would therefore require a replacement for this Picone-based compactness step.
Remark 16 (A nonsymmetric coupling). The assumptions above do not require the Jacobian \(f_u\) to be symmetric. For instance, for \(m=2\) let \[f_1(x,u)=b(x)\bigl((u^1+u^2)^{\gamma-1} +\varepsilon (u^2)^{\gamma-1}\bigr), \qquad f_2(x,u)=b(x)(u^1+u^2)^{\gamma-1},\] where \(2<\gamma<2^*\), \(b\in C(\overline{\Omega})\), \(b\ge b_0>0\), and \(0<\varepsilon\ll1\). Then \[f_{1,u^2}-f_{2,u^1} = \varepsilon(\gamma-1)b(x)(u^2)^{\gamma-2},\] so \(f_u\) is genuinely nonsymmetric.
The growth, cooperativity, strict coupling, and large-growth assumptions are immediate. Since the nonlinearities are homogeneous of degree \(\gamma-1\), Euler’s identity gives \[\sum_{j=1}^2 f_{i,u^j}(x,u)u^j=(\gamma-1)f_i(x,u), \qquad i=1,2,\] and hence the radial monotonicity condition holds. Moreover, with \(s=u^1+u^2\), \[Q_f(x,u)\ge(\gamma-1)b(x)s^\gamma, \qquad f(x,u)\cdot u \le b(x)(1+\varepsilon M_\gamma)s^\gamma,\] where \(M_\gamma:=\max_{0\le t\le1}(1-t)t^{\gamma-1}\). Thus \[Q_f(x,u) \ge \frac{\gamma-1}{1+\varepsilon M_\gamma}\,f(x,u)\cdot u .\] Hence, if \(0<\varepsilon<(\gamma-2)/M_\gamma\), then 14 holds with some \(\theta>1\). Therefore the minimax scheme applies to this genuinely non-variational system.
We now specialize the abstract framework to a one-dimensional elliptic system. Let \(\Omega=(0,1)\), \(m\ge1\), and consider \[\label{fd1} \left\{ \begin{align} -u^k_{xx} &=\lambda (u^k)^{q-1}+f_k(x,u), && x\in(0,1),\quad k=1,\ldots,m,\\ u^k&\ge0, && x\in(0,1),\quad k=1,\ldots,m,\\ u^k&=0, && x=0,1,\quad k=1,\ldots,m . \end{align} \right.\tag{16}\] Here \(u=(u^1,\ldots,u^m)\), \(1<q<2\), and \(\mathcal{L}=-d^2/dx^2\). We assume that the nonlinearities \(f_k\) satisfy the assumptions stated in Section 5; for simplicity, in 11 , 12 we take \(\gamma_1=\gamma_2=:\gamma\).
Set \[\mathbb{W}:=(H_0^1(0,1))^m, \qquad g(u):=\bigl((u^1)^{q-1},\ldots,(u^m)^{q-1}\bigr)^T .\] We use the scalar cone \[S^o := \left\{ w\in H_0^1(0,1)\cap C([0,1]): w>0 \;\text{in }(0,1) \right\},\] and the product cones \(\mathbb{S}^o:=(S^o)^m\), \(\mathbb{S}:=\overline{\mathbb{S}^o}^{\,\mathbb{W}}\). The regular outer cone is \(\mathbb{U}^o:=(U^o)^m\), where \[U^o := \left\{ w\in S^o: w''\in L^\infty_{\rm loc}(0,1), \quad -w''\ge0 \text{ in }\mathcal{D}'(0,1) \right\}.\] Since every positive concave function in \(H_0^1(0,1)\) satisfies a Hopf-type lower bound near the boundary, this definition is compatible with the regular cone used in Section 5. Moreover, every \(w\in U^o\) is concave in \((0,1)\).
For \(u\in\mathbb{S}^o\) and \(v\in\mathbb{S}\setminus\{0\}\), define \[\mathcal{R}(u,v) = \frac{ \displaystyle \sum_{k=1}^m\int_0^1 (u^k)'(v^k)'\,dx - \langle f(\cdot,u),v\rangle_m }{ \displaystyle \langle g(u),v\rangle_m }.\] Here and below, \[\langle h(\cdot,u),v\rangle_m := \sum_{k=1}^m\int_0^1 h_k(x,u)v^k\,dx .\] The corresponding minimax level is \[\lambda^* := \sup_{u\in\mathbb{U}^o} \inf_{v\in\mathbb{S}\setminus\{0\}} \mathcal{R}(u,v).\] By Lemma 11, \(\lambda^*<+\infty\). Moreover, \(\lambda^*>0\). Indeed, let \(\omega\in H_0^1(0,1)\) be the torsion function \[-\omega''=1,\qquad \omega>0 \;\text{in }(0,1),\] and set \(\boldsymbol{\omega}:=(\omega,\ldots,\omega)\). The same estimate as in the proof of Lemma 11 gives, for sufficiently small \(t_0>0\), \[\mathcal{R}(t_0\boldsymbol{\omega},v)\ge c>0 \qquad \forall v\in\mathbb{S}\setminus\{0\}.\] Hence \(\lambda^*\ge c>0\). For \(u\in\mathbb{U}^o\), set \[\lambda(u):= \inf_{v\in\mathbb{S}\setminus\{0\}}\mathcal{R}(u,v).\]
Lemma 17. The lower consistency condition \({\rm (LC)}\) is satisfied for 16 .
Proof. Choose \(\delta_0\in(0,\lambda^*)\), and let \(u\in\mathbb{U}^o\) satisfy \(\lambda(u)\ge\lambda^*-\delta_0\). Then \(\lambda(u)>0\). We prove that \[\liminf_{r\to\infty} \inf_{v\in\mathbb{S}_r\setminus\{0\}} \mathcal{R}(\mathbb{I}_ru,v) \ge\lambda(u), \qquad \mathbb{I}_ru:=(I_ru^1,\ldots,I_ru^m).\] Since each \(u^k\) is concave, its nodal interpolant satisfies \(0\le I_ru^k\le u^k\) in \((0,1)\). Hence, because \(1<q<2\), \[g(\mathbb{I}_ru)\le g(u) \quad\text{componentwise}, \qquad \langle g(\mathbb{I}_ru),v\rangle_m \le \langle g(u),v\rangle_m\] for every \(v\in\mathbb{S}_r\setminus\{0\}\).
Let \(v\in\mathbb{S}_r\setminus\{0\}\). Since \(v\) is piecewise linear and \(I_r\) is the nodal interpolant, integration over each mesh interval gives \[\sum_{k=1}^m\int_0^1 (I_ru^k)'(v^k)'\,dx = \sum_{k=1}^m\int_0^1 (u^k)'(v^k)'\,dx .\] Moreover, by \(f_{i,u^j}\ge0\) and \(\mathbb{I}_ru\le u\), we have \(f_i(x,\mathbb{I}_ru)\le f_i(x,u)\), \(i=1,\ldots,m\). Since \(v\ge0\), it follows that \[\langle\mathcal{A}(\mathbb{I}_ru),v\rangle_m \ge \langle\mathcal{A}(u),v\rangle_m .\] By the definition of \(\lambda(u)\), we have \[\langle\mathcal{A}(u),v\rangle_m \ge \lambda(u)\langle\mathcal{G}(u),v\rangle_m \qquad \forall v\in\mathbb{S}_r\setminus\{0\}.\] Combining the preceding inequalities and using \(\lambda(u)>0\), we obtain \[\langle\mathcal{A}(\mathbb{I}_ru),v\rangle_m \ge \lambda(u)\langle\mathcal{G}(\mathbb{I}_ru),v\rangle_m .\] Since the denominator is positive on \(\mathbb{S}_r\setminus\{0\}\), this gives \[\mathcal{R}(\mathbb{I}_ru,v)\ge\lambda(u) \qquad \forall v\in\mathbb{S}_r\setminus\{0\}.\] Taking the infimum over \(v\) and then the lower limit as \(r\to\infty\), we obtain \({\rm (LC)}\). ◻
Theorem 18. Assume that system 16 satisfies the assumptions stated above. Then 16 admits a maximal one-sided saddle-node point \[(u^*,\lambda^*)\in\mathbb{U}^o\times(0,+\infty).\] Moreover, 16 has no weak solution in \(\mathbb{U}^o\) for \(\lambda>\lambda^*\). Furthermore, there exists \(v^*\in\mathbb{S}\setminus\{0\}\) such that \[\label{MainBd1} \lambda^* = \mathcal{R}(u^*,v^*) = \inf_{v\in\mathbb{S}\setminus\{0\}}\mathcal{R}(u^*,v) = \sup_{u\in\mathbb{U}^o} \inf_{v\in\mathbb{S}\setminus\{0\}} \mathcal{R}(u,v).\qquad{(7)}\] In addition, \[\mathcal{F}(u^*,\lambda^*)=0 \quad\text{in }\mathbb{W}^*, \qquad \bigl\langle D_u\mathcal{F}(u^*,\lambda^*)\xi,v^* \bigr\rangle=0 \quad \forall \xi\in\mathbb{W} .\] If, moreover, \(v^*\in\mathbb{S}^o\), then \[\lambda^* = \min_{v\in\mathbb{S}\setminus\{0\}}\mathcal{R}(u^*,v) = \min_{v\in\mathbb{S}^o}\mathcal{R}(u^*,v).\]
Proof. By Lemmas 11, 12, and 13, all assumptions of Theorem 5 are satisfied except possibly the lower consistency condition \({\rm (LC)}\). The latter is proved in Lemma 17. Hence Theorem 5 applies and yields a pair \[(u^*,v^*)\in\mathbb{U}^o\times(\mathbb{S}\setminus\{0\})\] and a number \(\lambda^*>0\) such that ?? holds, together with \[\mathcal{F}(u^*,\lambda^*)=0 \quad\text{in }\mathbb{W}^*, \qquad \bigl\langle D_u\mathcal{F}(u^*,\lambda^*)\xi,v^* \bigr\rangle=0 \quad \forall \xi\in\mathbb{W} .\] Therefore \((u^*,\lambda^*)\) is a maximal one-sided saddle-node point in the cone \(\mathbb{U}^o\).
The nonexistence of weak solutions in \(\mathbb{U}^o\) for \(\lambda>\lambda^*\) follows from the minimax characterization. Indeed, if \(\mathcal{F}(u,\lambda)=0\) for some \(u\in\mathbb{U}^o\), then, by the denominator positivity, \[\mathcal{R}(u,v)=\lambda \qquad \forall v\in\mathbb{S}\setminus\{0\}.\] Hence \[\lambda = \inf_{v\in\mathbb{S}\setminus\{0\}}\mathcal{R}(u,v) \le \lambda^* .\]
If \(v^*\in\mathbb{S}^o\), then \(v^*\) is admissible in the inner problem over \(\mathbb{S}^o\). Since \[\mathcal{R}(u^*,v^*) = \inf_{v\in\mathbb{S}\setminus\{0\}}\mathcal{R}(u^*,v),\] and \(\mathbb{S}^o\subset\mathbb{S}\setminus\{0\}\), we obtain \[\min_{v\in\mathbb{S}\setminus\{0\}}\mathcal{R}(u^*,v) = \mathcal{R}(u^*,v^*) = \min_{v\in\mathbb{S}^o}\mathcal{R}(u^*,v).\] ◻
We have shown that, under suitable compactness, positivity, and approximation assumptions, the maximal one-sided saddle-node value of 3 is characterized by the minimax formula 2 . The construction is developed for abstract nonlinear equations and then verified for a class of elliptic systems, including non-variational ones.
Theorems 4 and 14 show that the Galerkin scheme produces finite - dimensional maximal one-sided saddle-node points and that their limits are singular solutions \((u^*,\hat{\lambda}^*)\). Under the stronger assumptions of Theorem 5, the limiting value coincides with the maximal one-sided saddle-node value. Thus the method gives a direct way to locate the critical level without first constructing a whole solution branch.
In finite dimension, the formula becomes \[\lambda_r(u) = \min_{1\le i\le N_r} \frac{\langle\mathcal{A}(u),\eta_i\rangle}{\langle\mathcal{G}(u),\eta_i\rangle}, \qquad \lambda_r^* = \max_{u\in\mathcal{S}_r^o}\lambda_r(u),\] where \(\{\eta_1,\ldots,\eta_{N_r}\}\) is the positive basis of the Galerkin cone. Hence the computation of \(\lambda_r^*\) is reduced to a finite-dimensional nonsmooth maximization problem. This may be viewed as a nonlinear Collatz–Wielandt type formula and treated by standard tools of nonsmooth optimization [22]–[24]. Related finite-dimensional minimax formulas have proved useful in nonlinear bifurcation and eigenvalue problems [11]–[13], [15].
The convergence results are close in spirit to finite-element eigenvalue approximation, going back to Courant and developed further in [25]–[27]. Here, however, the approximated object is not a linear eigenvalue, but a nonlinear one-sided saddle-node value selected by a minimax procedure. In this sense, the perturbation estimate of Theorem 10 plays the role of a Rayleigh-quotient type stability bound, in analogy with classical perturbation arguments for spectral problems [28].
Formula 2 is also related to variational characterizations of principal spectral quantities in ordered settings, including the Birkhoff–Varga principle for essentially positive matrices [29] and its extensions [16]. For elliptic operators, related characterizations of principal eigenvalues appear in the works of Donsker–Varadhan [30], Protter–Weinberger [31], Nussbaum–Pinchover [32], and Berestycki–Nirenberg–Varadhan [33].
The applications considered here are partly motivated by concave–convex problems, where scalar equations are often treated by ordered sub- and supersolutions; see, for instance, [34], [35]. For systems, this order structure is considerably more restrictive. The present minimax formula offers a different route: it identifies the critical one-sided saddle-node value directly from the extended Rayleigh quotient.
Several questions remain open. It would be natural to extend the method to broader classes of nonlinear and non-variational systems, possibly under weaker assumptions on the finite-dimensional cones. In particular, an important problem is whether the convergence of finite-dimensional minimax bifurcation formulas to their infinite-dimensional counterpart can be obtained by different methods and under less restrictive hypotheses.
Finally, the proposed method should be viewed as complementary to classical bifurcation theory. The Crandall–Rabinowitz theorem, Krasnosel’skiı̆’s theory, and Rabinowitz’s global theorem describe the local or global structure of solution sets under spectral and degree-theoretic assumptions. By contrast, the present approach provides a direct minimax formula for the critical one-sided saddle-node value. Under additional nondegeneracy assumptions, classical local theory can then be used to prove that the singular point selected by the minimax formula is indeed a genuine bifurcation point.
This appendix records a finite-dimensional realization of the cone, quotient-compatibility, and \(M\)-matrix assumptions required in the Galerkin construction of Theorem 4. The Hopf-type boundary estimates used in the continuous elliptic problem are imposed separately.
Let \(\Omega\subset\mathbb{R}^d\) be a bounded domain, and set \[W:=H_0^1(\Omega), \qquad \mathcal{C}^0:=C(\overline{\Omega}).\] We use the positive cone \[S^o := \left\{ v\in W\cap C(\overline{\Omega}): v=0 \;\text{on }\partial\Omega,\quad v>0 \;\text{in }\Omega \right\}, \qquad S:=\overline{S^o}^{\,\mathcal{C}^0}\cap W .\]
Let \(\{\Omega_r\}_{r\ge1}\) be polyhedral subdomains such that \(\Omega_r\subset\Omega\) and \[\bigcup_{r\ge1}\Omega_r=\Omega .\] The index \(r\) includes both the choice of the subdomain \(\Omega_r\) and the mesh size. If \(\Omega\) itself is polyhedral, this construction reduces to the usual finite element approximation on the fixed domain: one takes \(\Omega_r=\Omega\) for all \(r\), and \(r\) denotes only the mesh refinement parameter.
Let \(\mathcal{T}_r\) be a shape-regular conforming \(P_1\)-triangulation of \(\Omega_r\), and let \(W_r\) be the corresponding finite element space with zero boundary values on \(\partial\Omega_r\), extended by zero to \(\Omega\). Thus \(W_r\subset H_0^1(\Omega)\). We assume the Galerkin density property: for every \(\xi\in W\) there exist \(\xi_r\in W_r\) such that \[\xi_r\to\xi \qquad\text{strongly in }W .\] This is standard for interior polyhedral exhaustions and shape-regular finite element spaces; see, for example, [26], [36], [37].
Let \(B_1,\ldots,B_{N_r}\) be the interior nodes of \(\mathcal{T}_r\), and let \(\{\psi_1,\ldots,\psi_{N_r}\}\) be the associated nodal basis. Then \[\psi_i(B_j)=\delta_{ij}, \qquad \psi_i\ge0, \qquad \psi_i=0 \quad\text{on }\partial\Omega_r .\] After extension by zero outside \(\Omega_r\), each \(\psi_i\) belongs to \(H_0^1(\Omega)\cap C(\overline{\Omega})\).
Define the nodal cones \[S_r^o := \left\{ u_r=\sum_{i=1}^{N_r}u_i\psi_i\in W_r: u_i>0,\;i=1,\ldots,N_r \right\},\] and \[S_r:=\overline{S_r^o}^{\,W_r} = \left\{ u_r=\sum_{i=1}^{N_r}u_i\psi_i\in W_r: u_i\ge0,\;i=1,\ldots,N_r \right\}.\] Then \(S_r^o\) is a nonempty open subset of \(W_r\), and \(S_r\) is a closed polyhedral cone with positive basis \(\{\psi_1,\ldots,\psi_{N_r}\}\).
Moreover, \(S_r\subset S\). Indeed, every \(u_r\in S_r\) is a nonnegative continuous function in \(H_0^1(\Omega)\) and vanishes on \(\partial\Omega\). If \(\phi\in S^o\) is fixed, then \(u_r+\varepsilon\phi\in S^o\) for every \(\varepsilon>0\), and \(u_r+\varepsilon\phi\to u_r\) in \(C(\overline{\Omega})\) as \(\varepsilon\downarrow0\). Hence \(u_r\in S\).
Thus the cones \(S_r^o\) and \(S_r\) satisfy the abstract cone condition \({\rm (C)}\), namely \({\rm (c1)}\)–\({\rm (c2)}\). Notice that, when \(\Omega_r\subsetneq\Omega\), one does not generally have \(S_r^o\subset S^o\), since functions in \(W_r\) vanish outside \(\Omega_r\). This is precisely why the discrete denominator positivity is included in \({\rm (CQ)}\).
For \(u\in C(\overline{\Omega})\), define the nodal interpolant \[\mathcal{I}_r u:=\sum_{i=1}^{N_r}u(B_i)\psi_i .\] If \(u\in S^o\), then \(B_i\in\Omega\) and \(u(B_i)>0\) for every \(i=1,\ldots,N_r\). Therefore \(\mathcal{I}_r u\in S_r^o\). Hence the interpolation condition \({\rm (c3)}\) holds.
We next verify the discrete denominator positivity for the concave term used in the elliptic system. Let \(1<q<2\) and \(g(u)=u^{q-1}\). If \(u_r\in S_r^o\) and \(v_r\in S_r\setminus\{0\}\), then \[\int_\Omega u_r^{q-1}v_r\,dx>0 .\] Indeed, writing \[u_r=\sum_{i=1}^{N_r}u_i\psi_i, \qquad v_r=\sum_{i=1}^{N_r}v_i\psi_i,\] we have \(u_i>0\), \(v_i\ge0\), and \(v_{i_0}>0\) for at least one index \(i_0\). Since \(\psi_{i_0}>0\) on a set of positive measure and \(u_r\ge u_{i_0}\psi_{i_0}\), the integrand \(u_r^{q-1}v_r\) is positive on a set of positive measure. Thus the denominator part of \({\rm (CQ)}\) holds in the scalar case.
Similarly, if \(u\in S^o\) and \(v\in S\setminus\{0\}\), then \[\int_\Omega u^{q-1}v\,dx>0 .\] Indeed, \(v\ge0\), \(v\not\equiv0\), and \(u>0\) in \(\Omega\). Thus the continuous denominator condition \({\rm (D)}\) holds for the concave term.
For systems, we use the product spaces and cones \[\mathbb{W}_r:=(W_r)^m, \qquad \mathbb{S}_r^o:=(S_r^o)^m, \qquad \mathbb{S}_r:=(S_r)^m .\] Then \(\mathbb{S}_r\subset\mathbb{S}\), and \(\mathbb{S}_r\) is a polyhedral cone with basis \[\{\psi_i e_k:\;i=1,\ldots,N_r,\;k=1,\ldots,m\},\] where \(e_k\) is the \(k\)-th coordinate vector in \(\mathbb{R}^m\). The componentwise interpolant satisfies \[\mathbb{I}_r u\in\mathbb{S}_r^o \qquad \text{for every }u\in\mathbb{S}^o .\] Moreover, for \(u_r\in\mathbb{S}_r^o\) and \(v_r\in\mathbb{S}_r\setminus\{0\}\), \[\langle g(u_r),v_r\rangle_m = \sum_{k=1}^m \int_\Omega (u_r^k)^{q-1}v_r^k\,dx >0 .\] Thus the product cones satisfy \({\rm (C)}\), and the denominator part of \({\rm (CQ)}\) follows. The finite-dimensional \(C^1\)-regularity required in \({\rm (CQ)}\) is immediate for the restrictions considered in the elliptic applications.
Let \(a(\cdot,\cdot)\) be the scalar bilinear form used in the elliptic problem, for instance the form in 10 . Set \[A_r=(a_{ij})_{i,j=1}^{N_r}, \qquad a_{ij}:=a(\psi_j,\psi_i).\] If \(u_r=\sum_j u_j\psi_j\), then \[(A_r\mathbf{u})_i=a(u_r,\psi_i), \qquad \mathbf{u}=(u_1,\ldots,u_{N_r})^T .\] We assume that \(A_r\) satisfies the strong discrete maximum principle \[\label{eq:impl} A_r\vartheta\ge0,\quad \vartheta\ne0 \quad\Longrightarrow\quad \vartheta>0 .\tag{17}\] This holds, for example, when \(A_r\) is a nonsingular irreducible \(M\)-matrix. Sufficient mesh conditions are classical; see [26], [29], [36], [38]. In dimension \(d=1\), the standard piecewise-linear discretization of \(-d^2/dx^2\) on any partition satisfies 17 .
Finally, we record the discrete comparison principle used in the compactness argument.
Lemma 19 (Discrete comparison for a sublinear problem). Let \(A_r\) be a nonsingular \(M\)-matrix, \(0<p<1\), and \(c_0>0\). Suppose that \(U,W\in S_r^o\), written in nodal coordinates, satisfy \[A_rU\ge c_0U^p, \qquad A_rW=c_0W^p ,\] where the powers are understood componentwise. Then \(U\ge W\).
Proof. Suppose that \(U\not\ge W\), and set \[\tau:=\max\{s\in(0,1):sW\le U\}.\] Then \(0<\tau<1\), and for some \(i_0\), \[U_{i_0}=\tau W_{i_0}, \qquad U_i-\tau W_i\ge0,\quad i=1,\ldots,N_r .\] Put \(Z:=U-\tau W\). Since \(Z\ge0\), \(Z_{i_0}=0\), and \(A_r\) has nonpositive off-diagonal entries, we have \[(A_rZ)_{i_0}\le0 .\] On the other hand, \[(A_rZ)_{i_0} = (A_rU-\tau A_rW)_{i_0} \ge c_0U_{i_0}^p-\tau c_0W_{i_0}^p = c_0(\tau^p-\tau)W_{i_0}^p>0,\] because \(0<p<1\). This contradiction proves \(U\ge W\). ◻
In the system case, Lemma 19 is applied componentwise, with the same stiffness matrix \(A_r\) in each component.
We record the Picone-type inequalities used in the compactness arguments.
Lemma 20 (Continuous Picone inequality). Let \[\mathcal{L} u = -\partial_{x_i}\bigl(\sigma_{ij}(x)u_{x_j}\bigr)+c(x)u,\] where \((\sigma_{ij})\) is symmetric and uniformly elliptic. Let \(u\ge0\), \(v>0\) in \(\Omega\), and assume that \(u,v\in H_0^1(\Omega)\) and \(u^2/v\in H_0^1(\Omega)\). Then \[\langle\mathcal{L} u,u\rangle \ge \left\langle \mathcal{L} v,\frac{u^2}{v} \right\rangle .\]
Proof. It is enough to prove the identity for smooth positive functions; the general case follows by approximation. The zero-order terms cancel. For the principal part, \[\sigma_{ij}v_{x_j} \left(\frac{u^2}{v}\right)_{x_i} - \sigma_{ij}u_{x_j}u_{x_i} = - \sigma_{ij} \left(u_{x_j}-\frac{u}{v}v_{x_j}\right) \left(u_{x_i}-\frac{u}{v}v_{x_i}\right) \le0 .\] After integration over \(\Omega\), this gives \[\left\langle \mathcal{L} v,\frac{u^2}{v} \right\rangle - \langle\mathcal{L} u,u\rangle \le0,\] and the assertion follows. ◻
Lemma 21 (Discrete Picone inequality). Let \(A=(a_{ij})_{i,j=1}^r\) be symmetric and satisfy \(a_{ij}\le0\) for \(i\ne j\). Then, for every \(\mathbf{u}\in\mathbb{R}_+^r\) and \(\mathbf{v}\in(0,\infty)^r\), \[\mathbf{u}^T A\mathbf{u} \ge (A\mathbf{v})\cdot\frac{\mathbf{u}^2}{\mathbf{v}}, \qquad \frac{\mathbf{u}^2}{\mathbf{v}} := \left( \frac{u_1^2}{v_1},\ldots,\frac{u_r^2}{v_r} \right).\]
Proof. Set \(z_i:=u_i/v_i\). By symmetry, \[\mathbf{u}^T A\mathbf{u} - (A\mathbf{v})\cdot\frac{\mathbf{u}^2}{\mathbf{v}} = -\frac{1}{2} \sum_{i,j=1}^r a_{ij}v_iv_j(z_i-z_j)^2 .\] The diagonal terms vanish, and \(a_{ij}\le0\) for \(i\ne j\). Hence the right-hand side is nonnegative. ◻
The componentwise system version is obtained by summing Lemma 21 over the components. Symmetry and the nonpositive off-diagonal sign condition are essential in the discrete Picone identity; an \(M\)-matrix property alone, without symmetry, is not sufficient.
We record the estimate which justifies the linearization of \(u^{q-1}\), \(1<q<2\), on the regular positive cone. The Hopf-type lower bound in the definition of \(\mathbb{U}^o\) compensates for the boundary singularity of the derivative.
Lemma 22 (Boundedness of the linearized concave term). Let \(1<q<2\), and set \[g(u):=\bigl((u^1)^{q-1},\ldots,(u^m)^{q-1}\bigr)^T .\] For every \(u\in\mathbb{U}^o\), the formal derivative \[\bigl\langle Dg(u)\xi,\eta\bigr\rangle_m = (q-1)\sum_{k=1}^m \int_\Omega (u^k)^{q-2}\xi^k\eta^k\,dx\] defines a bounded operator \(Dg(u):\mathbb{W}\to\mathbb{W}^*\). In particular, \(D_u\mathcal{F}(u,\lambda)\) is well defined for every \(u\in\mathbb{U}^o\) and \(\lambda\in\mathbb{R}\).
Proof. Since \(u\in\mathbb{U}^o\), each component satisfies \(u^k\ge c_kd_\Omega\) in \(\Omega\). Hence, because \(1<q<2\), \[(u^k)^{q-2}\le C d_\Omega^{q-2}\le C d_\Omega^{-1}.\] By the Hardy inequality, \[\int_\Omega (u^k)^{q-2}|\xi^k||\eta^k|\,dx \le C \left\|\frac{\xi^k}{d_\Omega}\right\|_{L^2} \|\eta^k\|_{L^2} \le C\|\xi^k\|_{H_0^1}\|\eta^k\|_{H_0^1}.\] Summing over \(k=1,\ldots,m\) gives \[\bigl| \langle Dg(u)\xi,\eta\rangle_m \bigr| \le C\|\xi\|_{\mathbb{W}}\|\eta\|_{\mathbb{W}}.\] The assertion follows. ◻
We verify the closedness properties used in the passage from Galerkin saddle-point pairs to the limiting elliptic system. Throughout this appendix, functions defined on \(\Omega_r\) are extended by zero to \(\Omega\). We use that the admissible domains \(\Omega_r\Subset\Omega\) satisfy a uniform Hardy inequality and exhaust \(\Omega\) from inside.
Lemma 23 (Residual and linearized closedness). Let \(r_n\to\infty\), \(\lambda_n\to\lambda\), \(u_n\in\mathbb{S}_{r_n}^o\), \(v_n\in\mathbb{S}_{r_n}^o\), and assume that \[u_n\to u \quad\text{strongly in }\mathbb{W}, \qquad v_n\rightharpoonup v \quad\text{weakly in }\mathbb{W}, \qquad u,v\in\mathbb{S}\setminus\{0\}.\] Then, for every \(\zeta\in\mathbb{W}\) and every \(\zeta_n\in\mathbb{W}_{r_n}\) such that \(\zeta_n\to\zeta\) strongly in \(\mathbb{W}\), \[\langle\mathcal{F}(u_n,\lambda_n),\zeta_n\rangle \to \langle\mathcal{F}(u,\lambda),\zeta\rangle .\]
Assume, in addition, that \(u\in\mathbb{U}^o\) and that \((u_n)\) satisfies the discrete barrier estimate \[u_n^k\ge c\,d_{\Omega_{r_n}} \qquad\text{in }\Omega_{r_n},\quad k=1,\ldots,m,\] with \(c>0\) independent of \(n\), where \(d_{\Omega_{r_n}}(x):=\operatorname{dist}(x,\partial\Omega_{r_n})\). Then, for every \(\xi\in\mathbb{W}\) and every \(\xi_n\in\mathbb{W}_{r_n}\) such that \(\xi_n\to\xi\) strongly in \(\mathbb{W}\), \[\bigl\langle D_u\mathcal{F}(u_n,\lambda_n)\xi_n,v_n \bigr\rangle \to \bigl\langle D_u\mathcal{F}(u,\lambda)\xi,v \bigr\rangle .\]
Proof. The residual convergence follows from the continuity of \(a_m\), the strong convergence \(u_n\to u\) in \(\mathbb{W}\), compact subcritical embeddings, and the growth assumptions on \(f\) and \(g\).
We prove the convergence of the linearized identities. The terms \(a_m(\xi_n,v_n)\) and \(\langle f_u(u_n)\xi_n,v_n\rangle_m\) pass to the limit by the strong convergence of \(\xi_n\), the weak convergence of \(v_n\), compact subcritical embeddings, and the growth assumption on \(f_u\). It remains to treat \[\sum_{k=1}^m \int_{\Omega_{r_n}} (u_n^k)^{q-2}\xi_n^k v_n^k\,dx .\] Fix \(K\Subset\Omega\). For all large \(n\), \(K\subset\Omega_{r_n}\), and the interior exhaustion gives \(d_{\Omega_{r_n}}\ge\delta_K>0\) on \(K\). Hence the barrier estimate implies \(u_n^k\ge c\delta_K\) on \(K\). Since \(1<q<2\), the functions \((u_n^k)^{q-2}\) are uniformly bounded on \(K\) and converge a.e. to \((u^k)^{q-2}\). Therefore \[(u_n^k)^{q-2}\xi_n^k \to (u^k)^{q-2}\xi^k \quad\text{strongly in }L^2(K).\] As \(v_n^k\rightharpoonup v^k\) weakly in \(L^2(K)\), we get \[\int_K (u_n^k)^{q-2}\xi_n^k v_n^k\,dx \to \int_K (u^k)^{q-2}\xi^k v^k\,dx .\]
It remains to control the boundary part. By the discrete barrier estimate, \(1<q<2\), and the uniform Hardy inequality, \[\int_E (u_n^k)^{q-2}|\xi_n^k||v_n^k|\,dx \le C\|\xi_n^k\|_{H_0^1(\Omega_{r_n})}\|v_n^k\|_{L^2(E)}\] for every measurable \(E\subset\Omega_{r_n}\). Taking \(E=\Omega_{r_n}\setminus K\), using the compactness of the embedding \(\mathbb{W}\hookrightarrow\mathbb{L}^2\), which gives \(v_n\to v\) strongly in \(\mathbb{L}^2\), and then choosing \(K\uparrow\Omega\), we make this contribution uniformly small. The same Hardy estimate, now using the Hopf-type lower bound for \(u\), controls the limiting integral on \(\Omega\setminus K\). Hence \[\int_{\Omega_{r_n}} (u_n^k)^{q-2}\xi_n^k v_n^k\,dx \to \int_\Omega (u^k)^{q-2}\xi^k v^k\,dx .\] Summing over \(k=1,\ldots,m\) and using \(\lambda_n\to\lambda\), we obtain the desired convergence. ◻
Remark 24. In the extended \((PS)_e\)-sequences used in Lemma 12, the required discrete barrier estimate is obtained by comparison with the positive scalar sublinear Galerkin solution. Hence Lemma 23 verifies the linearized closedness condition \({\rm (CD)}\) for the elliptic Galerkin limits needed in the proof of Theorem 14.