June 10, 2026
In this paper we consider the Dirichlet boundary value problem driven by the weighted Laplacian equation and considered on the Sierpiński gasket. Under suitable growth conditions imposed on the nonlinear term, we provide existence and multiplicity results employing the direct method, mountain pass theorem and some related multiplicity result.
Michał Bełdziński
Institute of Mathematics, Lodz University of Technology,
al. Politechniki 8, 93-590 Łódź, Poland
michal.beldzinski@p.lodz.pl
Krzysztof Jelito
Institute of Mathematics, Lodz University of Technology,
al. Politechniki 8, 93-590 Łódź, Poland
Department of Mathematics, Silesian University of Technology,
Kaszubska 23, 44-100 Gliwice, Poland
krzysztof.jelito@dokt.p.lodz.pl
Krzysztof.Jelito@polsl.pl
The authors contributed equally to this work.
Keywords. Sierpiński gasket, weighted Laplacian, variational methods, Mountain Pass Lemma, multiplicity results
2020 Mathematics Subject Classification. 28A80, 35J50, 35J92
We study a Dirichlet boundary value problem involving a weighted Laplacian on the Sierpiński gasket \(V\), namely \[\label{equ:Main95inclusion} \begin{cases} -\Delta_a u(\boldsymbol{x}) \ni f\big(\boldsymbol{x},u(\boldsymbol{x})\big) & \text{for } \boldsymbol{x}\in V\setminus V_0,\\[2mm] u(\boldsymbol{x})=0 & \text{for } \boldsymbol{x}\in V_0, \end{cases}\tag{1}\] where \(f\colon V\times\mathbb{R}\to\mathbb{R}\) is an \(L^1\)–Carathéodory function satisfying suitable growth assumptions specified below, function \(a\) is bounded and strictly positive. \(V_0\) denotes the intrinsic boundary of \(V\). The passage from equality to inclusion is necessitated by the fact that, the operator \(-\Delta_a\) may be multivalued. A detailed explanation of this phenomenon will be discussed in Section 2.3.
Differential equations on fractal domains have attracted considerable attention in recent decades. We refer to [1]–[3] for background material on analysis on fractals and related elliptic problems.
Falconer and Hu [4] established the existence of solutions to the problem \[\begin{cases} \Delta u + a(\boldsymbol{x})u = f\big(\boldsymbol{x},u(\boldsymbol{x})\big) & \text{for \mu-a.e. } \boldsymbol{x}\in V\setminus V_0,\\ u(\boldsymbol{x})=0 & \text{for } \boldsymbol{x}\in V_0, \end{cases}\] where \(a\) is integrable and the nonlinearity \(f\) satisfies suitable growth conditions near zero and at infinity. Their approach relied on variational methods, in particular the Mountain Pass and Saddle Point theorems.
Subsequently, Molica Bisci and Rădulescu [5], as well as Molica Bisci, Bonanno, and Rădulescu [6], proved existence and multiplicity results for various generalizations of the problem originally studied by Falconer and Hu.
In [7], [8], Galewski investigated the boundary value problem \[\begin{cases} \Delta u(\boldsymbol{x}) + a(\boldsymbol{x})u(\boldsymbol{x}) = f\big(\boldsymbol{x},u(\boldsymbol{x}),w(\boldsymbol{x})\big) + g(\boldsymbol{x}) & \text{for \mu-a.e. } \boldsymbol{x}\in V\setminus V_0,\\ u(\boldsymbol{x})=0 & \text{for } \boldsymbol{x}\in V_0, \end{cases}\] with \(a,g\in L^1(V)\), \(w\in L^2(V)\), and a nonlinearity \(f\) satisfying appropriate growth assumptions. In [7] the case \(g\equiv0\) was considered and the dependence of solutions on parameters was analyzed via the Mountain Pass Theorem, while in [8] monotonicity methods were employed to establish existence and uniqueness results in the fractal setting.
More recently, Priyadarshi and Sahu [9] studied systems of equations involving the \(p\)–Laplacian on fractal domains.
It seems that boundary value problems involving weighted Laplacians on fractals have not yet been addressed in the literature. The definition of the weighted Laplacian proposed in the present paper allows us to work within the Hilbert space \(H_0^1(V)\), which is particularly convenient for variational methods.
Our main tool in establishing the existence of nontrivial solutions is a set of abstract results from [10]. These theorems combine the direct minimization method with the Mountain Pass Lemma. To the best of the authors’ knowledge, they have not previously been applied to equations on the Sierpiński gasket \(V\), even in the case of the standard Laplacian \(-\Delta\). In particular, the present work appears to be the first to employ them in the study of equations driven by the weighted Laplacian considered here.
The paper is organized as follows. In Section 2, we establish basic properties of Szulkin-type functionals together with the methods applied. Then, we recall the standard construction of the Sierpiński gasket via an iterated function system, following [1]. Next, we introduce the Laplacian \(-\Delta\) in the sense of Kigami [2], [11] and subsequently define the weighted Laplacian \(-\Delta_a\), emphasizing its relation to the classical Laplacian on \(V\). Then, we define a Szulkin-type functional that plays the role of the Euler action functional associated with 1 . We impose assumptions on the nonlinearity \(f\) and derive a series of lemmas establishing the key properties of the corresponding nonlinear term. This leads to the formulation of the final Szulkin-type functional, whose properties are further investigated. Finally, in Section 3 we apply these methods to problem 1 .
Throughout this subsection we assume that \[\label{ass:E}\boxed{ (E,\|\cdot\|)\;\text{is a real and reflexive Banach space.} }\tag{2}\] The norm in the dual space \(E^*\) is denoted by \(\|\cdot\|_{E^*}\). Moreover, for every \(r>0\) we set \[B_r := \{u\in E : \|u\|<r\}.\] On the space \(E\) we consider a Szulkin-type functional of the form \[\label{equ:Szulkin-type95functional} I = \psi + \Phi,\tag{3}\] where \[\label{ass:psi}\boxed{ \psi \colon E \to \mathbb{R}\;\text{is convex and continuous.} }\tag{4}\] It is common to assume that the functional \(\psi\) is lower semicontinuous. In the present setting, however, where \(\psi\) takes only finite values, this assumption is in fact equivalent to continuity; see [12]. The same applies to weak lower semicontinuity, see [13]. Recall that, for every \(u\in E\), the functional \(\psi\) admits a subdifferential given by \[\partial \psi(u) = \left\{\eta\in E^* : \psi(v)-\psi(u) \ge \langle \eta, v-u\rangle \text{ for all } v\in E\right\}.\] Concerning \(\Phi\), we shall assume that \[\label{ass:Phi}\boxed{ \begin{tabular}{l} \Phi \colon E \to \mathbb{R} is of class C^1. Moreover, \Phi' is weak-to-strong continuous, i.e.\\ \qquad u_n \rightharpoonup u in E implies \Phi'(u_n) \to \Phi'(u) in E^*. \end{tabular} }\tag{5}\] Following [14], we call \(u\in E\) a critical point of \(I\) if \(-\Phi'(u)\in \partial\psi(u)\), or equivalently, if \[\psi(v)-\psi(u) + \langle \Phi'(u), v-u\rangle \ge 0 \quad \text{for all } v\in E.\]
We say that \(I\) satisfies the Palais–Smale condition if every sequence \((u_n)\subset E\) such that:
the sequence \(\bigl(I(u_n)\bigr)\) is bounded,
there exists a sequence \((\varepsilon_n)\subset \mathbb{R}\) with \(\varepsilon_n\to 0\) and \[\label{equ:almost95critical} \psi(v)-\psi(u_n)+\langle \Phi'(u_n), v-u_n\rangle \ge -\varepsilon_n \|v-u_n\| \quad \text{for all } v\in E,\tag{6}\]
admits a convergent subsequence. Sequences satisfying the above conditions are called Palais–Smale sequences for \(I\).
We impose the following additional assumption on \(\psi\): \[\label{ass:psi95rho}\boxed{ \begin{tabular}{l} there exists an increasing function \rho\colon[0,\infty)\to[0,\infty) with \rho(0)=0 s.t.\\ \qquad \psi(v)\ge \psi(u)+\langle \eta, v-u\rangle +\rho(\|u-v\|)\|u-v\|\\ for all u,v\in E and every \eta\in \partial\psi(u). \end{tabular} }\tag{7}\]
It was shown in [15] that condition 7 is equivalent to the \(\rho\)-uniform monotonicity of \(\partial\psi\) and to the so-called \(\rho\)-uniform convexity of \(\psi\). Both notions are discussed in detail in [15]. Here we restrict ourselves to assumption 7 , which allows us to prove the following counterpart of [16].
Lemma 1. Assume that 2 , 4 , 7 , and 5 hold. Then the functional \(I\) defined by 3 satisfies the Palais–Smale condition if and only if every Palais–Smale sequence possesses a bounded subsequence.
Proof. The necessity is obvious. We show sufficiency. As observed in [14], condition 6 is equivalent to the existence of a sequence \((\eta_n)\subset E^*\) such that \(\eta_n\to 0\) and \[\eta_n - \Phi'(u_n) \in \partial\psi(u_n).\] Let \((u_n)\) be a bounded Palais–Smale sequence. Passing to a subsequence if necessary (not relabeled), we may assume that \(u_n\rightharpoonup u\) in \(E\). Choosing \(v=u\) in 6 , and \(v=u_n\) in 7 with an arbitrary \(\eta\in \partial\psi(u)\), we obtain \[\label{equ:prop:every} \rho(\|u_n-u\|)\|u_n-u\| \le \varepsilon_n\|u-u_n\| + \langle \Phi'(u_n), u-u_n\rangle + \langle \eta, u-u_n\rangle .\tag{8}\] Since \(u_n\rightharpoonup u\), the sequence \((\|u-u_n\|)\) is bounded and hence \(\varepsilon_n\|u-u_n\|\to 0\). By assumption 5 , we have \(\Phi'(u_n)\to \Phi'(u)\) in \(E^*\), which implies \[\bigl|\langle \Phi'(u_n), u-u_n\rangle\bigr| \le \|\Phi'(u_n)-\Phi'(u)\|_{E^*}\|u-u_n\| + |\langle \Phi'(u), u-u_n\rangle| \to 0.\] Moreover, \(\langle \eta, u-u_n\rangle \to 0\). Therefore, 8 yields \(\|u_n-u\|\to 0\), proving the claim. ◻
Lemma 2. Assume that 2 , 4 , 7 , and 5 hold, and define \(I\) by 3 . Then \(I\) is bounded from below on bounded subsets of \(E\).
Proof. We show that both \(\psi\) and \(\Phi\) are bounded from below on bounded sets. For \(\psi\) this follows from [15]. To verify boundedness of \(\Phi\), recall (see [12]) that \[\Phi(u)=\int_0^1 \langle \Phi'(tu),u\rangle\,dt \quad \text{for all } u\in E.\] Fix \(r>0\) and set \[M := \sup_{u\in B_r}\|\Phi'(u)\|_{E^*}.\] To show that \(M\) is finite let \((u_n)\subset B_r\) be such that \(\|\Phi'(u_n)\|_{E^*}\to M\). Passing to a subsequence, we may assume that \(u_n\rightharpoonup u\) for some \(u\in B_r\). By 5 , we have \(M=\|\Phi'(u)\|_{E^*}<\infty\). Consequently, \[|\Phi(u)| \le \int_0^1 \|\Phi'(tu)\|_{E^*}\|u\|\,dt \le Mr \quad \text{for all } u\in B_r,\] which completes the proof. ◻
Proposition 1. Assume that 2 , 4 , 7 , and 5 hold, and define \(I\) by 3 . If \(I\) is coercive, i.e. \[\lim_{\|u\|\to\infty} I(u)=\infty,\] then \(I\) satisfies the Palais–Smale condition. Moreover, \(I\) attains its global minimum: there exists \(u^*\in E\) such that \[I(u^*)=\min_{u\in E} I(u).\]
Proof. Let \((u_n)\) be a Palais–Smale sequence for \(I\). By coercivity, boundedness of \(\bigl(I(u_n)\bigr)\) implies boundedness of \((u_n)\). Lemma 1 then shows that \(I\) satisfies the Palais–Smale condition. Moreover, coercivity yields the existence of \(R>0\) such that \[I(0)<\inf_{u\in E\setminus B_R} I(u) \quad \text{and} \quad I(0)<\inf_{u\in \partial B_R} I(u).\] The conclusion now follows from [10]. ◻
The following result links [14] with [10], both of which rely on the mountain pass geometry. In particular, it guarantees the existence of two critical points: one corresponding to a local minimum of \(I\) on a sufficiently small ball, and another given by the saddle point provided by the Mountain Pass Lemma.
Proposition 2. Assume that 2 , 4 , 7 , and 5 hold, and define \(I\) by 3 . Suppose further that \(I\) satisfies the Palais–Smale condition and that there exist \(r>0\) and \(e\in E\setminus \overline{B_r}\) such that \[\label{equ:Mountain95Pass95geometry95for95I} I(0) < \inf_{u\in \partial B_r} I(u) \quad \text{and} \quad I(e)\le I(0).\qquad{(1)}\] Then \(I\) admits at least one nonzero critical point. If, in addition, \[\label{equ:condition95for95third95critical95point} \inf_{u\in \overline{B_r}} I(u) < I(0),\qquad{(2)}\] then \(I\) possesses at least two nonzero critical points.
Proof. Since \(I\) satisfies the Palais–Smale condition, Szulkin’s version of the Mountain Pass Lemma applies; see [14]. This yields the existence of a nonzero critical point. If both ?? and ?? hold, then Lemma 2 allows us to apply [10], which completes the proof. ◻
Let \(\boldsymbol{p}_1\), \(\boldsymbol{p}_2\), and \(\boldsymbol{p}_3\) be the vertices of an arbitrary equilateral triangle in \(\mathbb{R}^2\). Each point \(\boldsymbol{p}_i\), \(i\in\{1,2,3\}\), is the unique fixed point of the affine contraction \(\boldsymbol{F}_i\colon \mathbb{R}^2\to\mathbb{R}^2\) defined by \[\boldsymbol{F}_i(\boldsymbol{x})=\frac{\boldsymbol{x}+\boldsymbol{p}_i}{2} \quad \text{for all } \boldsymbol{x}\in\mathbb{R}^2.\] Hence, \(\mathbb{R}^2\) equipped with the Euclidean norm, together with the mappings \(\boldsymbol{F}_1\), \(\boldsymbol{F}_2\), and \(\boldsymbol{F}_3\), constitutes an iterated function system; see [1]. Let \(\mathcal{K}(\mathbb{R}^2)\) denote the family of all compact subsets of \(\mathbb{R}^2\), endowed with the Hausdorff metric \[h(X,Y) = \max\Bigl\{ \max_{\boldsymbol{x}\in X}\min_{\boldsymbol{y}\in Y}|\boldsymbol{x}-\boldsymbol{y}|, \max_{\boldsymbol{y}\in Y}\min_{\boldsymbol{x}\in X}|\boldsymbol{x}-\boldsymbol{y}| \Bigr\} \quad\text{for all } X,Y\in\mathcal{K}(\mathbb{R}^2).\] Define the mapping \(\mathcal{F}\colon \mathcal{K}(\mathbb{R}^2)\to\mathcal{K}(\mathbb{R}^2)\) by \[\mathcal{F}(X)=\boldsymbol{F}_1(X)\cup \boldsymbol{F}_2(X)\cup \boldsymbol{F}_3(X)\quad \text{for every } X\in\mathcal{K}(\mathbb{R}^2).\] Then \(\mathcal{F}\) is a contraction on \(\bigl(\mathcal{K}(\mathbb{R}^2),h\bigr)\) and therefore admits a unique fixed point, known as the Sierpiński gasket, which we denote by \(V\). Setting \[V_0=\{\boldsymbol{p}_1,\boldsymbol{p}_2,\boldsymbol{p}_3\},\] and defining recursively \[V_{n+1}=\mathcal{F}(V_n) \quad\text{for every } n\in\mathbb{N}_0,\] we obtain \(h(V_n,V)\to 0\) as \(n\to\infty\). This construction of \(V\) was used to define the Laplacian in [17] and [18]. As in [4], we equip \(V\) with the normalized Hausdorff measure \(\mu\), so that \(\mu(V)=1\).
We introduce two standard notational conventions concerning points in \(V\). For simplicity, we identify ordered \(n\)-tuples with entries in \(\{1,2,3\}\) with words of length \(n\) over the same alphabet. Accordingly, we set \({\Sigma_n=\{1,2,3\}^n}\) and \(\Sigma_\infty=\bigcup_{n\in\mathbb{N}_+}\Sigma_n\). For \(\omega=(\omega_1,\dots,\omega_n)\in\Sigma_n\) we define \[\boldsymbol{F}_\omega = \boldsymbol{F}_{\omega_1}\circ\cdots\circ \boldsymbol{F}_{\omega_n},\] and \[\boldsymbol{p}_\omega = \boldsymbol{F}_{\omega_1\ldots\omega_{n-1}}(\boldsymbol{p}_{\omega_n}).\] Consequently, \[V_{n-1}=\{\boldsymbol{p}_\omega : \omega\in\Sigma_n\} \quad \text{for every } n\in\mathbb{N}_+.\]
We now introduce the Laplace operator on \(V\), following [17]. The starting point is the observation that for every \(u\in C(V)\) one has \[\label{equ:fundamental95inequality} \sum_{1\le i<j\le 3}\bigl|u(\boldsymbol{p}_i)-u(\boldsymbol{p}_j)\bigr|^2 \le \tfrac{5}{3}\sum_{k=1}^3\sum_{1\le i<j\le 3} \bigl|u(\boldsymbol{p}_{ki})-u(\boldsymbol{p}_{kj})\bigr|^2.\tag{9}\] This inequality is fundamental in the construction of the operator \(\Delta\) on \(V\). Informally, it states that the energy determined by the values of \(u\) on \(V_0\) does not exceed the energy determined on \(V_1\), provided the latter is rescaled by the factor \(\tfrac{5}{3}\). As a consequence, one obtains \[\label{equ:condition95E9440n4195is95increasing} \sum_{1\le i<j\le 3}\bigl|u(\boldsymbol{p}_{\omega i})-u(\boldsymbol{p}_{\omega j})\bigr|^2 \le \tfrac{5}{3}\sum_{k=1}^3\sum_{1\le i<j\le 3} \bigl|u(\boldsymbol{p}_{\omega ki})-u(\boldsymbol{p}_{\omega kj})\bigr|^2,\tag{10}\] for every \(\omega\in\Sigma_\infty\) and arbitrary \(u\in C(V)\), see [3] for details. This leads to the definition of the energy functional \[\label{equ:functional95E} \mathcal{E}(u) = \lim_{n\to\infty} \left(\tfrac{5}{3}\right)^n \sum_{\omega\in\Sigma_n} \sum_{1\le i<j\le 3} \bigl|u(\boldsymbol{p}_{\omega i})-u(\boldsymbol{p}_{\omega j})\bigr|^2.\tag{11}\] The limit (finite or infinite) exists, since the sequence is nondecreasing by 10 . Note that in defining \(\mathcal{E}(u)\) we only use the values of \(u\) on \[V_\infty=\bigcup_{n\in\mathbb{N}_+}V_n = \{\boldsymbol{p}_\omega : \omega\in\Sigma_\infty\}.\] It is therefore natural to restrict the definition of \(\mathcal{E}\) to continuous functions. We define \[H^1(V)=\{u\in C(V): \mathcal{E}(u)<\infty\},\] and equip this space with the norm \[\|u\|_{\mathcal{E}} = \left(\mathcal{E}(u)+\int_V |u(\boldsymbol{x})|^2\,d\mu(\boldsymbol{x})\right)^{1/2}.\] As an analogue of the space of functions vanishing on the boundary, we introduce \[H_0^1(V) = \{u\in H^1(V): u(\boldsymbol{x})=0 \text{ for all }\boldsymbol{x}\in V_0\},\] which, endowed with the norm \[\|u\|_{H_0^1(V)}=\sqrt{\mathcal{E}(u)},\] is a Hilbert space; see [19]. Within this framework, the Laplace operator is defined in the weak sense as \(-\Delta\colon H_0^1(V)\to \bigl(H_0^1(V)\bigr)^*\) given by \(-\Delta = \tfrac{1}{2} \mathcal{E}'\).
This definition coincides with Kigami’s weak Laplacian on the Sierpiński gasket [2]; see also [3]. It is consistent with the classical definition on a bounded open set \(\Omega\subset\mathbb{R}^n\), where \(-\Delta=\operatorname{div}\nabla\) is realized as the Gâteaux derivative of \[H_0^1(\Omega)\ni v\longmapsto \tfrac12\int_\Omega |\nabla v(\boldsymbol{x})|^2\,d\boldsymbol{x}.\] It can be shown (see [4]) that, as in the classical case, solutions to \[\begin{cases} -\Delta u(\boldsymbol{x})=f(\boldsymbol{x}), & \boldsymbol{x}\in V\setminus V_0,\\ u(\boldsymbol{x})=0, & \boldsymbol{x}\in V_0, \end{cases}\] with \(f\in L^2(V)\) are precisely the critical points of the functional \[H_0^1(V)\ni v\longmapsto \tfrac12 \mathcal{E}(v) - \int_V f(\boldsymbol{x})v(\boldsymbol{x})\,d\mu(\boldsymbol{x}).\]
Another analogy with the classical setting is provided by the following Sobolev-type embedding.
Lemma 3 ([20]). The embedding \(H^1(V)\hookrightarrow C(V)\) is compact. Moreover, \[\|u\|_{\infty}\le 9\,\|u\|_{H_0^1(V)} \quad \text{for all } u\in H_0^1(V).\]
Since \(C(V)\) embeds continuously into \(L^2(V)\), it follows that \(H_0^1(V)\) embeds compactly into \(L^2(V)\). Let \(\lambda_1\) denote the first eigenvalue of the Laplace operator with Dirichlet boundary conditions. Its reciprocal provides the optimal constant in the Poincaré inequality \[\label{equ:Poincare95inequality} \|u\|_{L^2}^2 \le \lambda_1^{-1}\|u\|_{H_0^1}^2 \quad \text{for all } u\in H_0^1(V).\tag{12}\] Unlike the one-dimensional case, no explicit formula for \(\lambda_1\) is known. Numerical approximations based on finite element methods yield \(\lambda_1\approx 16.816\); see [11].
We begin by recalling the definition of the weighted Laplacian \(-\Delta_a\) in the classical setting, that is, on an open and bounded subset \(\Omega\) of Euclidean space. This serves as a point of reference for the construction that follows. Building on the resulting analogies–while emphasizing the essential differences–we then proceed to define the operator \(-\Delta_a\) on the Sierpiński gasket.
Let us recall that the Laplace operator \(-\Delta\) in its weak formulation–namely, as a mapping from \(H^1_0(\Omega)\) into the dual space \(H^{-1}(\Omega)\)–is defined as the Gâteaux derivative of the functional \[\label{equ:potential95of95Delta} H^1_0(\Omega) \ni v \longmapsto \tfrac{1}{2}\int_\Omega |\nabla v(\boldsymbol{x})|^2\, d\boldsymbol{x}.\tag{13}\] Given a function \(a\colon \Omega \to \mathbb{R}\) that is measurable, bounded, and bounded away from zero, we obtain the corresponding potential for the operator \(-\Delta_a\) in the form \[\label{equ:potential95of95Delta95a} H^1_0(\Omega) \ni v \longmapsto \tfrac{1}{2}\int_\Omega a(\boldsymbol{x})\,|\nabla v(\boldsymbol{x})|^2\, d\boldsymbol{x}.\tag{14}\] In the case of the Sierpiński gasket, the potential of the Laplacian is given by \(\tfrac{1}{2}\mathcal{E}\), where the energy form \(\mathcal{E}\) is defined by 11 . It is important to note that the definition of \(\mathcal{E}(u)\) relies exclusively on the values of \(u\) on the set \(V_\infty\), which has measure zero. Consequently, one cannot unambiguously define the operator \(-\Delta_a\) using an expression of the form \[\label{equ:functional95E95a95for95continuous} \mathcal{E}_a(u) = \lim_{n\to \infty} \left(\tfrac{5}{3}\right)^{n} \sum_{\omega \in \Sigma_n} \sum_{1 \leq i < j \leq 3} \tfrac{a(\boldsymbol{p}_{\omega i}) + a(\boldsymbol{p}_{\omega j})}{2}\big|u(\boldsymbol{p}_{\omega i})-u(\boldsymbol{p}_{\omega j})\big|^2.\tag{15}\] for an arbitrary function \(a\in L^\infty(V)\). However, it turns out that the above definition can indeed be employed in the case of continuous functions, since \(V_\infty\) is dense in \(V\). Additional properties of the functional \(\mathcal{E}_a\) are described in the following
Lemma 4. Take a continuous function \(a\colon V\longrightarrow (0,\infty)\). Then, for every \(u\in H^1_0(V)\), there exists a limit \[\label{equ:functional95E95a95for95continuous952} \mathcal{E}_a(u) = \lim_{n\to \infty} \left(\tfrac{5}{3}\right)^{n} \sum_{\omega \in \Sigma_n} \sum_{1 \leq i < j \leq 3} \tfrac{a(\boldsymbol{p}_{\omega i}) + a(\boldsymbol{p}_{\omega j})}{2}\big|u(\boldsymbol{p}_{\omega i})-u(\boldsymbol{p}_{\omega j})\big|^2.\tag{16}\] Moreover \(\mathcal{E}_a\), defined above, is of class \(C^1\) and \((u,v) \longmapsto \langle\mathcal{E}_a'(u),v\rangle\) is an inner product on \(H^1_0(V)\), equivalent with the standard one.
Proof. Let us denote \[a_\omega = \min_{\boldsymbol{x} \in F_\omega V} a(\boldsymbol{x}), \quad a_- = \min_{\boldsymbol{x}\in V} a(\boldsymbol{x})\quad \text{and}\quad a_+ = \max_{\boldsymbol{x}\in V} a(\boldsymbol{x}).\] Then we immediately get \[\label{equ:direct95inequalities} {a_-}\mathcal{E}(u) \leq \mathcal{E}_a(u) \leq {a_+}\mathcal{E}(u)\quad \text{for every }u\in H^1_0(V).\tag{17}\] Moreover, for any word \(\omega \in \Sigma_\infty\) and every \(k \in \{1,2,3\}\), we clearly have \(a_{\omega} \leq a_{\omega k}\) since \(\boldsymbol{F}_{\omega k}V \subset \boldsymbol{F}_\omega V\). Using condition 10 , for any \(n\in \mathbb{N}\), we have \[\label{equ:non-decreasing95E95a} \begin{align} \left(\tfrac{5}{3}\right)^{n} \sum_{\omega \in \Sigma_n} \sum_{1 \leq i < j \leq 3} a_\omega \big|u(\boldsymbol{p}_{\omega i})-u(\boldsymbol{p}_{\omega j})\big|^2 & \leq \left(\tfrac{5}{3}\right)^{n} \sum_{\omega \in \Sigma_n} \tfrac{5}{3} \sum_{k = 1}^3 \sum_{1 \leq i < j \leq 3} a_\omega \big|u(\boldsymbol{p}_{\omega k i})-u(\boldsymbol{p}_{\omega k j})\big|^2\\ & \leq \left(\tfrac{5}{3}\right)^{n + 1} \sum_{\omega \in \Sigma_n} \sum_{k = 1}^3 \sum_{1 \leq i < j \leq 3} a_{\omega k} \big|u(\boldsymbol{p}_{\omega k i})-u(\boldsymbol{p}_{\omega k j})\big|^2\\ & = \left(\tfrac{5}{3}\right)^{n + 1} \sum_{\sigma \in \Sigma_{n+1}} \sum_{1 \leq i < j \leq 3} a_{\sigma} \big|u(\boldsymbol{p}_{\sigma i})-u(\boldsymbol{p}_{\sigma j})\big|^2 \end{align}\tag{18}\] The last equality is achieved by taking \(\sigma = \omega k\). By the following inequality \[\left|a_\omega - a(\boldsymbol{p}_{\omega i})\right| \leq \sup\left\{|a(\boldsymbol{x}) - a(\boldsymbol{y})| : \boldsymbol{x}, \boldsymbol{y}\in V,\;|\boldsymbol{x} - \boldsymbol{y}| \leq \sqrt{3} |\boldsymbol{p}_1 - \boldsymbol{p}_2|2^{-n - 1}\right\}\] and by uniform continuity of \(a\), we get \[\sup_{\substack{\omega \in \Sigma_n \\ 1\leq i < j \leq 3}}\left|\tfrac{a(\boldsymbol{p}_{\omega i}) + a(\boldsymbol{p}_{\omega j})}{2} - a_\omega\right| \to 0 \quad \text{as }n\to \infty.\] Hence we also get that \[\left|\left(\tfrac{5}{3}\right)^{n} \sum_{\omega \in \Sigma_n} \sum_{1 \leq i < j \leq 3} \left(\tfrac{a(\boldsymbol{p}_{\omega i}) + a(\boldsymbol{p}_{\omega j})}{2} - a_\omega\right)\big|u(\boldsymbol{p}_{\omega i})-u(\boldsymbol{p}_{\omega j})\big|^2\right|\] is bounded above by a term \[\sup_{\substack{\omega \in \Sigma_n \\ 1\leq i < j \leq 3}}\left|\tfrac{a(\boldsymbol{p}_{\omega i}) + a(\boldsymbol{p}_{\omega j})}{2} - a_\omega\right| \mathcal{E}(u)\] that converges to \(0\) as \(n\to \infty\) and consequently we can decompose \[\label{equ:n-th95part95of95E95a} \left(\tfrac{5}{3}\right)^{n} \sum_{\omega \in \Sigma_n} \sum_{1 \leq i < j \leq 3} \tfrac{a(\boldsymbol{p}_{\omega i}) + a(\boldsymbol{p}_{\omega j})}{2}\big|u(\boldsymbol{p}_{\omega i})-u(\boldsymbol{p}_{\omega j})\big|^2\tag{19}\] as a sum of two terms: \[\begin{align} \left(\tfrac{5}{3}\right)^{n} \sum_{\omega \in \Sigma_n} \sum_{1 \leq i < j \leq 3} & a_\omega\big|u(\boldsymbol{p}_{\omega i})-u(\boldsymbol{p}_{\omega j})\big|^2\\ & + \left(\tfrac{5}{3}\right)^{n} \sum_{\omega \in \Sigma_n} \sum_{1 \leq i < j \leq 3} \left(\tfrac{a(\boldsymbol{p}_{\omega i}) + a(\boldsymbol{p}_{\omega j})}{2} - a_\omega\right)\big|u(\boldsymbol{p}_{\omega i})-u(\boldsymbol{p}_{\omega j})\big|^2 \end{align}\] where one is non-decreasing by 18 and clearly bounded by \(a_+\mathcal{E}(u)\) The second one converges to \(0\). This shows that the desired limit exists. Hence one can pass to the limit with the parallelogram law described for 19 to get \[\mathcal{E}_a(u - v) + \mathcal{E}_a(u + v) = 2\mathcal{E}_a(u) + 2 \mathcal{E}_a(v).\] Now it is clear that \(\mathcal{E}_a\) is a square of a norm introducing an inner product on \(H^1_0(V)\), that is equivalent with the standard one by 17 . Consequently all desired properties follows. ◻
To allow discontinuous weight \(a\), the construction of the Laplacian on \(V\) forces us to define \(a\) on the set \(V_\infty\). Accordingly, we impose the following assumption \[\label{ass:a}\boxed{ \begin{tabular}{l} \text{function a\colon V_\infty \longrightarrow (0, \infty) satisfy} \\ \qquad 0 < {a_-} = \inf\limits_{\boldsymbol{x}\in V_\infty} a(\boldsymbol{x}) \leq \sup\limits_{\boldsymbol{x}\in V_\infty} a(\boldsymbol{x}) = {a_+} < \infty \end{tabular} }\tag{20}\] and define \(\mathcal{E}_a\colon H^1_0(V)\longrightarrow \mathbb{R}\), as a natural generalization of 14 , by the formula \[\label{equ:functional95E95a} \mathcal{E}_a(u) = \limsup_{n\to \infty}\left(\tfrac{5}{3}\right)^{n} \sum_{\omega \in \Sigma_n} \sum_{1 \leq i < j \leq 3} \tfrac{a(\boldsymbol{p}_{\omega i}) + a(\boldsymbol{p}_{\omega j})}{2}\big|u(\boldsymbol{p}_{\omega i})-u(\boldsymbol{p}_{\omega j})\big|^2\tag{21}\] for every \(u\in H^1_0(V)\). Note that, for every continuous function \(a\colon V \longrightarrow (0,\infty)\), the restriction \(a|_{V_\infty}\) satisfies 20 . Therefore, the functional \(\mathcal{E}_a\) given by 21 generalizes the weak Laplacian described in Lemma 4. In general, it is not known whether \(\mathcal{E}_a\) is of class \(C^1\). However, the estimates in 17 still hold. Some further properties of \(\mathcal{E}_a\) are described in Proposition 3.
Proposition 3. Assume that 20 holds. Then the functional \(\psi\colon H_0^1(V)\longrightarrow\mathbb{R}\) defined by \[\label{equ:psi} \psi(u) = \tfrac{1}{2}\mathcal{E}_a(u) \quad \text{for every }u\in H^1_0(V)\qquad{(3)}\] with \(\mathcal{E}_a\) given by 21 , satisfies 4 and 7 with \(\rho(x) = {a_-} x\).
Proof. We apply [21] to prove condition 7 , which is equivalent to convexity of \({\frac{1}{2}\mathcal{E}_a(\cdot) -\frac{{a_-}}{2}\|{\cdot}\|_{H_0^1(V)}^2}\). Note that \(\psi - \frac{{a_-}}{2}\mathcal{E}\) is convex since \[\psi(u) -\tfrac{{a_-}}{2}\mathcal{E}(u) = \tfrac12\limsup_{n\to \infty}\left(\tfrac{5}{3}\right)^{n} \sum_{\omega \in \Sigma_n} \sum_{1 \leq i < j \leq 3} \tfrac{\left(a(\boldsymbol{p}_{\omega i}) + a(\boldsymbol{p}_{\omega j})- 2 {a_-}\right)}{2}\big|u(\boldsymbol{p}_{\omega i})-u(\boldsymbol{p}_{\omega j})\big|^2.\] Indeed, for every fixed \(\boldsymbol{x},\boldsymbol{y}\in V\), the mapping \(u \longmapsto \frac{a(\boldsymbol{x}) + a(\boldsymbol{y}) - 2{a_-}}{2}\big|u(\boldsymbol{x})-u(\boldsymbol{y})\big|^2\) is convex. Hence each \[u\longmapsto \left(\tfrac{5}{3}\right)^{n} \sum_{\omega \in \Sigma_n} \sum_{1 \leq i < j \leq 3} \tfrac{a(\boldsymbol{p}_{\omega i}) + a(\boldsymbol{p}_{\omega j}) - 2{a_-}}{2}\big|u(\boldsymbol{p}_{\omega i})-u(\boldsymbol{p}_{\omega j})\big|^2\] defines a convex functional. Consequently \(\psi - \mathcal{E}_a\) is convex as an upper limit of convex functionals. Moreover \(\partial \left(\psi - \frac{{a_-}}{2}\mathcal{E}\right)(u) = \partial \psi(u) - \{ {a_-}\mathcal{E}'(u)\}\). Using that and the fact that a subdifferential of a convex functional is monotone we get \[\begin{align} \langle \eta - \xi, u - v\rangle \geq & {a_-} \langle \mathcal{E}'(u) - \mathcal{E}'(v), u - v\rangle = {a_-}\|u - v\|_{H^1_0}^2\\ & \text{for all }u,v\in H^1_0(V)\text{ and every }\eta \in \partial\psi(u), \xi \in \partial \psi(v), \end{align}\] which gives 7 by [15]. ◻
In general, the functional \(\mathcal{E}_a\) need not be Gâteaux differentiable. However, its convexity allows one to consider its subdifferential, in the sense of convex analysis, at every point. Consequently, as in the case of the weak Laplacian, for continuous \(a\) the operator \({-\Delta_a \colon H^1_0(V) \longrightarrow \left(H^1_0(V)\right)^*}\) is single-valued. When \(a\) is merely measurable, this no longer holds; in that case one obtains a maximally monotone operator \({-\Delta_a \colon H^1_0(V) \longrightarrow 2^{\left(H^1_0(V)\right)^*}}\). Recall that, for a fixed function \(f \in L^2(V)\), we will understand the inclusion \[\label{equ:inclusion95fixed95f} \begin{cases} -\Delta_a u(\boldsymbol{x}) \ni f(\boldsymbol{x}) & \text{for every }\boldsymbol{x}\in V\setminus V_0 \\ u(\boldsymbol{x}) = 0 & \text{for every }\boldsymbol{x}\in V_0 \end{cases}\tag{22}\] as \[\tfrac{1}{2}\mathcal{E}_a(v) \geq \tfrac{1}{2}\mathcal{E}_a(u) + \int_V f(\boldsymbol{x})\big(v(\boldsymbol{x}) - u(\boldsymbol{x})\big)\, d\mu(\boldsymbol{x}) \quad \text{for every }v\in H^1_0(V).\] Since \(\mathcal{E}_a\) is of class \(C^1\) under the assumption that \(a\) is continuous, we may consider, instead of the inclusion 22 , the equation \[\begin{cases} -\Delta_au(\boldsymbol{x}) = f(\boldsymbol{x}) & \text{for every }\boldsymbol{x}\in V\setminus V_0\\ u(\boldsymbol{x}) = 0 & \text{for every }\boldsymbol{x}\in V_0, \end{cases}\] In both cases, solutions are the critical points of the functional \[H^1_0(V) \ni v\longmapsto \tfrac{1}{2}\mathcal{E}_a(v) - \int_V f(\boldsymbol{x}) v(\boldsymbol{x})\, d\mu(\boldsymbol{x}).\] To consider \(f\) depending on \(u\), we impose suitable assumptions on \(f\).
Under the following, structural assumption on \(f\): \[\label{ass:f95Caratheodory} }} \boxed{ \begin{tabular}{l} \text{f\colon V\times \mathbb{R}\longrightarrow \mathbb{R} is \emph{an L^1-Carath\'eodory function}, that is:} \\ \quad \text{\bullet f(\boldsymbol{\cdot}, u) is measurable for every u\in \mathbb{R},} \\ \quad \text{\bullet f(\boldsymbol{x}, \cdot) is continuous for \mu-a.e. \boldsymbol{x}\in V,}\\ \quad \text{\bullet for every M > 0 there exists f_M\in L^1(V; \mathbb{R}_+) such that}\\ \quad \qquad \left|f(\boldsymbol{x}, u)\right| \leq f_M(\boldsymbol{x})\text{ for \mu-a.e. }\boldsymbol{x}\in V\text{ and every }u\in [-M,M]. \end{tabular} }\tag{23}\] we define an antiderivative of \(f(\boldsymbol{x}, \cdot)\) by the formula \[\label{equ:F-primitive95of95f} F(\boldsymbol{x},u) = \int_{0}^{u} f(\boldsymbol{x},s)\, ds\quad \text{for all }u\in \mathbb{R}\text{ and \mu-a.e. }\boldsymbol{x}\in V.\tag{24}\] The functional \(\Phi\) is defined in the following lemma.
Lemma 5. Assume that 23 holds. Then the functional \(\Phi\colon H_0^1(V)\longrightarrow \mathbb{R}\) defined by \[\label{equ:Phi} \Phi(u)= -\int_V F\big(\boldsymbol{x},u(\boldsymbol{x})\big)\, d\mu(\boldsymbol{x})\quad \text{for every }u\in H^1_0(V)\tag{25}\] satisfies 5 . Moreover \[\label{equ:Phi95derivative} \left\langle \Phi'(u), v\right\rangle = -\int_V f\big(\boldsymbol{x}, u(\boldsymbol{x})\big)v(\boldsymbol{x})\, d\mu(\boldsymbol{x})\quad \text{for every }u,v\in H^1_0(V).\tag{26}\]
Proof. Let \(u,v\in H_0^1(V)\). By 23 we have \[\lim\limits_{t\to0}\frac{F\big(\boldsymbol{x},u(\boldsymbol{x})+tv(\boldsymbol{x})\big)-F\big(\boldsymbol{x},u(\boldsymbol{x})\big)}{t}=f\big(\boldsymbol{x},u(\boldsymbol{x})\big)v(\boldsymbol{x})\quad \text{for \mu-a.e. }\boldsymbol{x}\in V.\] The embedding \(H_0^1(V) \hookrightarrow C(V)\) gives \(\|u\|_\infty , \|v\|_\infty \leq M\) for some \(M>0\) and hence the Lagrange mean value theorem yields \[\left|\frac{F\big(\boldsymbol{x},u(\boldsymbol{x})+tv(\boldsymbol{x})\big)-F\big(\boldsymbol{x},u(\boldsymbol{x})\big)}{t}\right|\le\max_{s\in[-2M,2M]}|f(\boldsymbol{x},s)|\cdot M \leq M f_{2M}(\boldsymbol{x})\] for \(\mu\)-a.e. \(\boldsymbol{x}\in V\) provided \(-1 \leq t \leq 1\). Using already mentioned imbedding \(H^1_0(V) \hookrightarrow C(V)\) together with [22] we instantly get continuity of \(\Phi'\) since for every \(u, v \in H^1_0(V)\) one has \[\begin{align} \left|\langle \Phi'(u), v\rangle\right| & = \int_V \left|f\big(\boldsymbol{x}, u(\boldsymbol{x})\big) v(\boldsymbol{x})\right|\, d\mu(\boldsymbol{x})\\ & \leq \|v\|_\infty \int_V \left|f\big(\boldsymbol{x}, u(\boldsymbol{x})\big) \right|\, d\mu(\boldsymbol{x}) \leq 9 \|v\|_{H^1_0}\left\|f\big(\boldsymbol{\cdot}, u(\boldsymbol{\cdot})\big)\right\|_{L^1} \end{align}\] by Lemma 3. ◻
We consider also the growth condition of the following form \[\label{ass:f95sublinear} } \boxed{ \begin{tabular}{l} \text{there exists \alpha < {a_-}\lambda_1 and \beta \in L^1(V; \mathbb{R}_+) such that}\\ \qquad F(\boldsymbol{x}, u) \leq \frac{\alpha}{2} |u|^2 + \beta(\boldsymbol{x})\text{ for all u\in \mathbb{R} and \mu-a.e. \boldsymbol{x}\in V} \end{tabular} }\tag{27}\] that provides the lower bound for \(\Phi\) of the following form.
Lemma 6. Assume that 23 and 27 hold. Then functional \(\Phi\) defined by 25 satisfies \[\Phi(u) \geq -\tfrac{\alpha}{2\lambda_1}\|u\|_{H^1_0}^2 - \|\beta\|_{L^1}\quad \text{for every }u\in H^1_0(V).\]
Proof. Using the Poincaré inequality 12 we get \[\begin{align} \Phi(u) & =-\int_VF(\boldsymbol{x},u(\boldsymbol{x}))\, d\mu\geq -\int_V \left( \tfrac{\alpha}{2}|u(\boldsymbol{x})|^2+\beta(\boldsymbol{x})\right)\, d\mu\\ &= -\tfrac{\alpha}{2}\|u\|_{L^2}^2-\|\beta\|_{L^1}\ge -\tfrac{\alpha}{2\lambda_1}\|u\|_{H_0^1}^2-\|\beta\|_{L^1}\quad \text{for every }u\in H^1_0(V).\quad\qedhere \end{align}\] ◻
Remark 1. Assumption 27 can be easily reformulated in terms of the function \(f\). It is sufficient to assume that there exist a constant \(\alpha < a_- \lambda_1\) and a function \(\beta \in L^1(V;\mathbb{R}_+)\) such that \[|f(\boldsymbol{x},u)| \le \alpha |u| + \beta(\boldsymbol{x}) \quad \text{for all } u \in \mathbb{R} \text{ and for \mu-a.e. } \boldsymbol{x}\in V .\] Moreover, the above assumption is in particular satisfied whenever there exist constants \(\alpha>0\), \(r\in(0,1)\) and a function \(\beta\in L^1(V;\mathbb{R}_+)\) such that \[f(\boldsymbol{x},u) \le \alpha |u|^{r} + \beta(\boldsymbol{x}) \quad \text{for all } u \in \mathbb{R} \text{ and for \mu-a.e. } \boldsymbol{x}\in V .\]
We consider the following assumption \[\label{ass:f950}\boxed{ \lim_{u \to 0} \frac{F(\boldsymbol{x}, u)}{|u|^2} = \infty \quad \text{uniformly with respect to \mu-a.e. }\boldsymbol{x}\in V }\tag{28}\] that allows to describe behaviour of \(\Phi\) near \(0\) in the following way.
Lemma 7. Assume that 23 , 28 hold and define \(\Phi\colon H^1_0(V) \longrightarrow \mathbb{R}\) using 25 . Then for all \(u\in H^1_0(V)\) and \(c > 0\) there exists \(t^* > 0\) such that \[\Phi(tu) < -c t^2\quad \text{for every }t\in (0,t^*).\]
Proof. Let \(\gamma = c\|u\|_{L^2}^{-2}\). By assumption 28 , there exists \(r > 0\) such that \[F(\boldsymbol{x}, u) \geq \gamma |u|^2\quad \text{for all }u\in [-r,r]\text{ and \mu-a.e. }\boldsymbol{x}\in V.\] Since \(H^1_0(V) \hookrightarrow C(V)\), we can take \(t^* = \min \left(r\|u\|_{\infty}^{-1},1\right)\). Then for any \(t\in (0,t^*)\) we get \(\|t u\|_\infty < r\) and hence \[\Phi(tu) = -\int_V F\big(\boldsymbol{x}, tu(\boldsymbol{x})\big)\, d\mu(\boldsymbol{x}) \leq -\int_V \gamma |tu(\boldsymbol{x})|^2\, d\mu(\boldsymbol{x}) = -\gamma t^2 \|u\|_{L^2}^2 = -c t^2\] for every \(t\in (0, t^*)\). ◻
Example 1. Taking \(f\colon V \times \mathbb{R} \longrightarrow \mathbb{R}\) given by \(f(\boldsymbol{x}, u) = |u|^{-\frac{1}{2}}u\) we get a standard nonlinearity satisfying 28 .
Let us recall that the Ambrosetti-Rabinovitz condition: \[\label{ass:f95Ambrosetti-Rabinowitz} }} \boxed{ \begin{tabular}{l} \text{there exist \theta > 2 and M_0 > 0 such that}\\ \qquad 0 < \theta F(\boldsymbol{x}, u) \leq uf(\boldsymbol{x}, u)\\ \text{for all u\in (-\infty, -M_0] \cup [M_0, \infty) and \mu-a.e. \boldsymbol{x}\in V} \end{tabular} }\tag{29}\] leads to the following lower bound for \(F\).
Lemma 8. Assume that 23 , 29 hold. Then there exist \(A>0\) and \(B\in L^1(V),\) such that \[F(\boldsymbol{x},u)\geq A|u|^\theta-B(\boldsymbol{x}) \quad \text{ for all u\in\mathbb{R} and \mu-a.e. \boldsymbol{x}\in V}.\]
Proof. We use standard arguments analogous to those in [23], adapted to the Sierpiński triangle \(V\). Let us start by setting \(M = \max\{1, M_0\}\) and fixing \(\boldsymbol{x} \in V\) for which 29 holds. Then we obtain the inequality \[\label{equ:lem32AR95consequence:proof} \frac{\theta}{v} \leq \frac{f(\boldsymbol{x},v)}{F(\boldsymbol{x},v)}, \quad \text{if } |v| \geq M.\tag{30}\] Integrating it over the interval \([M, u]\), with \(u > M\), with respect to \(v\), we get \(\theta \ln \frac{u}{M_0} \leq \ln \frac{F(\boldsymbol{x},u)}{F(\boldsymbol{x},M_0)}\). Hence, \[\label{equ:lem32AR95consequence:proof-2} A |u|^{\theta} \leq F(\boldsymbol{x},u)\tag{31}\] for \(u \geq M\), where \(A = \frac{F(\boldsymbol{x},M_0)}{M_0^{\theta}}\). Integrating 30 over \([v, -M]\), with \(v < -M\), shows that inequality 31 holds also for \(v \leq -M\). Finally, to obtain the assertion, it is enough to take \(B = f_M\), where \(f_M\) is the bound from assumption 23 . ◻
Together with 29 , it is typical to assume the following condition on \(F\) \[\label{equ:limsup95F610} \limsup_{u \to 0}\frac{F(\boldsymbol{x},u)}{|u|^2} = 0 \quad \text{uniformly for \mu-a.e. } \boldsymbol{x} \in V,\tag{32}\]
Example 2. Let \(f\colon V \times \mathbb{R} \longrightarrow \mathbb{R}\) be given by the formula \(f(\boldsymbol{x}, u) = |u|u\). Then \(F(\boldsymbol{x}, u) = \frac{1}{3}|u|^3\) and hence 32 holds. Assumption 29 holds as well for \(\theta = 3\) and any \(M_0 > 0\).
Condition 29 is clearly incompatible with 28 . Motivated by [10], we therefore propose an alternative assumption. Observe that whenever 32 holds, there exists \(R>0\) sufficiently small such that \[F(\boldsymbol{x},u) \le \tfrac{a_-|u|^2}{163} \le \tfrac{a_-R^2}{163} < \tfrac{a_-R^2}{162}\] for all \(u\in [-R,R]\) and for \(\mu\)-a.e. \(\boldsymbol{x} \in V\). This motivates the following assumption: \[\label{ass:f95R}\boxed{ \begin{tabular}{l} \text{there exist R>0 and \alpha<\frac{a_-R^2}{162} such that}\\ \qquad F(\boldsymbol{x},u)\le \alpha\\ \text{for all u\in[-R,R] and for \mu-a.e. \boldsymbol{x}\in V}, \end{tabular} }\tag{33}\] which is therefore satisfied. Observe that the seemingly unusual constant \(162\) is simply the product of the standard variational factor \(\frac{1}{2}\) and the square of the inverse of the constant \(9\) appearing in the embedding \(H^1(V) \hookrightarrow C(V)\). Consequently, the function \(f\) described in Example 2 satisfies both 32 and 29 . Below we provide an explicit example of a nonlinearity satisfying simultaneously 28 , 29 , and 33 .
Example 3. Consider, for simplicity, \(a \equiv 1\), and let \(f\colon V \times \mathbb{R}\longrightarrow \mathbb{R}\) be defined by \[f(\boldsymbol{x},u)= \begin{cases} \frac{1}{11} |u|^{-\frac{1}{2}}u &\text{ if } |u|\leq 100,\\ \frac{1}{11\,000} |u|u &\text{ if } |u|>100. \end{cases}\] Assumption 28 holds, since \(F(\boldsymbol{x}, u) = \frac{3}{22}|u|^\frac{3}{2}\) for \(u\in [-100,100]\) and \(\boldsymbol{x}\in V\). Hence \[\lim_{u\to 0}\frac{F(\boldsymbol{x}, u)}{|u|^2} = \lim_{u\to 0}\frac{3}{22\sqrt{|u|}} = \infty.\] If \(|u| > 100\) and \(\boldsymbol{x}\in V\), then \(F(\boldsymbol{x}, u) = \frac{|u|^3}{33\, 000} + \frac{1000}{33}\). To verify that 29 holds, we take \(\theta = \frac{23}{8} > 2\). The function \(F\) is positive for \(u\neq 0\). A straightforward computation shows that \(\theta F(\boldsymbol{x}, u) \leq u f(\boldsymbol{x}, u)\) is equivalent to \[\frac{1000}{33} \leq \frac{|u|^3}{24\cdot 11\, 000}\] for all \(u\in (-\infty, -100)\cup (100, \infty)\). Therefore, it suffices to take \(M_0 = 200\) in 29 . Next, since we assumed that \(a \equiv 1\), we have \(a_- = a_+ = 1\). Moreover, taking \(R = 100\) we obtain \[F(\boldsymbol{x}, u) \leq F(\boldsymbol{x}, R) = \frac{2000}{33} < 61 < \frac{10\, 000}{162} = \frac{a_- R^2}{162}\] for every \(\boldsymbol{x}\in V\) and \(u\in [-R,R]\). Consequently, 33 holds.
Assumption 33 allows to better control a geometry of \(\Phi\) as it is described in the following
Lemma 9. Assume that 23 , 33 hold and define \(\Phi \colon H^1_0(V)\longrightarrow \mathbb{R}\) using 25 . Then for every \(u\in H^1_0(V)\) with \(\|u\|_{H^1_0} = \frac{R}{9}\) we have \(\Phi(u) \geq -\alpha\).
Proof. Take \(u\) satisfying assumptions of the lemma. Then \(\|u\|_\infty \leq 9\|u\|_{H^1_0} = R\) and hence \[\Phi(u) = -\int_V F\big(\boldsymbol{x}, u(\boldsymbol{x})\big)\, d\mu(\boldsymbol{x}) \geq -\int_V \alpha\, d\mu(\boldsymbol{x}) = -\alpha. \qedhere\] ◻
Analogously to the case of the nonlinear Laplace equation with Dirichlet boundary condition described in [18], we call \(u\in H^1_0(V)\) a solution to the equation \[\label{equ:main95problem} \begin{cases} -\Delta_a u(\boldsymbol{x}) \ni f\big(\boldsymbol{x}, u(\boldsymbol{x})\big) & \text{for }\boldsymbol{x}\in V\setminus V_0,\\ u(\boldsymbol{x}) = 0 & \text{for }\boldsymbol{x}\in V_0, \end{cases}\tag{34}\] if the function is a critical point of the functional \(I\colon H^1_0(V) \longrightarrow \mathbb{R}\) defined by \[\label{equ:I} I(u) = \tfrac12\mathcal{E}_a(u)-\int_V F\big(\boldsymbol{x},u(\boldsymbol{x})\big)\, d\mu(\boldsymbol{x})\quad \text{for every }u\in H^1_0(V)\tag{35}\] with \(\mathcal{E}_a\) defined by 21 and \(F\) given by 24 . Equivalently \(u\) solves 34 , if the following hemivariational inequality holds \[\int_V f(\boldsymbol{x},u(\boldsymbol{x}))\big(v(\boldsymbol{x})-u(\boldsymbol{x})\big)\, d\mu(\boldsymbol{x}) +\tfrac12\mathcal{E}_a(v) \geq \tfrac12\mathcal{E}_a(u)\quad \text{for all }v\in H_0^1(V).\] Consequently \(I\) has the form 3 if we let \(\psi\) and \(\Phi\) be given by ?? and 25 , respectively. We begin with a basic result guaranteeing the solvability of equation 34 .
Theorem 4. Assume that 20 , 23 and 27 hold. Then problem 34 has at least one solution \(u^*\) satisfying \[\label{equ:minimization95I} I(u^*) = \min_{u\in H^1_0(V)}I(u)\tag{36}\] with \(I\) given by 35 .
Proof. We define \(I\) using formula 35 . By applying Proposition 3 and Lemma 5, we see that we are in a position to use Proposition 1 (assumption 2 holds trivially, since \(H^1_0(V)\) is a Hilbert space). Therefore, it suffices to show that \[\lim_{\|u\|_{H^1_0} \to \infty} I(u) = \infty.\] It follows immediately from estimates 17 together with Lemma 6. Indeed, \[I(u) \geq \tfrac{a_-}{2} \mathcal{E}(u) - \tfrac{\alpha}{2 \lambda_1} \|u\|_{H^1_0}^2 - \|\beta\|_{L^1} = \tfrac{1}{2} \left(a_- - \tfrac{\alpha}{\lambda_1}\right) \|u\|_{H^1_0}^2 - \|\beta\|_{L^1} \quad \text{for every } u \in H^1_0(V).\] The argument of a global minimum is always a critical point and hence, the existence of a solution is obtained. ◻
Before proceeding further, let us prove the following technical lemma, which, combined with Proposition 2, will allow us to establish the existence of two nontrivial solutions.
Lemma 10. Assume that 23 and 29 hold and let \(I\) be given by 35 . Then every Palais-Smale sequence \((u_n)\subset H_0^1(V)\) for the functional \(I\) is bounded.
Proof. Let \((u_n)\subset H_0^1(V)\) be the Palais-Smale sequence for \(I\). It means that there exists a sequence \((\varepsilon_n)\) satisfying \(\varepsilon_n\to 0\) and such that \[\psi(v)-\psi(u_n)+\langle\Phi'(u_n),v-u_n\rangle\geq -\varepsilon_n\|v-u_n\|_{H_0^1(V)}\quad \text{for all }v\in H_0^1(V),\] where \(\psi\) and \(\Phi\) are defined by ?? and 25 , respectively. In particular, taking \(v = (1 + t)u_n\), dividing both sides by \(t\) and passing to the limit \(t \downarrow 0\) we get \[\mathcal{E}_a(u_n) - \int_V f\big(\boldsymbol{x}, u_n(\boldsymbol{x})\big) u_n(\boldsymbol{x})\, d\mu(\boldsymbol{x}) \geq -\varepsilon_n\|u_n\|_{H_0^1}\quad \text{for every }n\in \mathbb{N}\] and hence \[\label{equ:proof95AR95property-1} -\tfrac{1}{\theta} \mathcal{E}_a(u_n) + \tfrac{1}{\theta} \int_V f\big(\boldsymbol{x}, u_n(\boldsymbol{x})\big) u_n(\boldsymbol{x})\, d\mu(\boldsymbol{x}) \leq \tfrac{\varepsilon_n}{\theta}\|u_n\|_{H_0^1}\quad \text{for every }n\in \mathbb{N}\tag{37}\] Now take \(M_0\) from assumption 29 . Then for every \(\boldsymbol{x}\in V\) either:
\(|u_n(\boldsymbol{x})| \geq M_0\) and then \[\tfrac{1}{\theta} f\big(\boldsymbol{x}, u_n(\boldsymbol{x})\big) u_n(\boldsymbol{x}) - F\big(\boldsymbol{x}, u_n(\boldsymbol{x})\big) \geq 0,\]
or \(|u_n(\boldsymbol{x})| < M_0\) and consequently, using function \(f_{M_0}\) from assumption 23 , we get \[\begin{align} \left|\tfrac{1}{\theta} f\big(\boldsymbol{x}, u_n(\boldsymbol{x})\big) u_n(\boldsymbol{x}) - F\big(\boldsymbol{x}, u_n(\boldsymbol{x})\big)\right| & \leq \tfrac{1}{\theta} \left| f\big(\boldsymbol{x}, u_n(\boldsymbol{x})\big) u_n(\boldsymbol{x}) \right| + \left| F\big(\boldsymbol{x}, u_n(\boldsymbol{x})\big)\right| \\ & \leq \tfrac{1}{\theta} f_{M_0}(\boldsymbol{x}) M_0 + \left|\int_0^{u(\boldsymbol{x})} f(\boldsymbol{x}, s)\, ds\right| \\ & \leq \tfrac{1 + \theta}{\theta} M_0 f_{M_0}(\boldsymbol{x}). \end{align}\]
Therefore, for every \(n\in \mathbb{N}\), we get \[\label{equ:proof95AR95property-2} \begin{align} \int_V \left(\tfrac{1}{\theta} f\big(\boldsymbol{x}, u_n(\boldsymbol{x})\big) u_n(\boldsymbol{x}) - F\big(\boldsymbol{x}, u_n(\boldsymbol{x})\big)\right)\, d\mu(\boldsymbol{x}) & \geq - \int_V \tfrac{1 + \theta}{\theta} M_0 f_{M_0}(\boldsymbol{x})\, d\mu(\boldsymbol{x})\\ & = -\tfrac{1 + \theta}{\theta} M_0 \|f_{M_0}\|_{L^1}. \end{align}\tag{38}\] Now adding \(I(u_n)\) to both sides of 37 gives \[\begin{align} I(u_n) + \tfrac{\varepsilon_n}{\theta}\|u_n\|_{H^1_0} & \geq \tfrac{\theta - 2}{2\theta} \mathcal{E}_a(u_n) + \int_V \left(\tfrac{1}{\theta} f\big(\boldsymbol{x}, u_n(\boldsymbol{x})\big) u_n(\boldsymbol{x}) - F\big(\boldsymbol{x}, u_n(\boldsymbol{x})\big)\right)\, d\mu(\boldsymbol{x}) \\ & \geq \tfrac{\theta - 2}{2\theta} \mathcal{E}_a(u_n) - \tfrac{1 + \theta}{\theta} M_0 \|f_{M_0}\|_{L^1} \\ & \geq a_- \tfrac{\theta - 2}{2\theta} \|u_n\|_{H^1_0}^2 - \tfrac{1 + \theta}{\theta} M_0 \|f_{M_0}\|_{L^1}\quad \text{for every }n\in \mathbb{N}. \end{align}\] As a consequence we obtain \[I(u_n) \geq a_- \tfrac{\theta - 2}{2\theta} \|u_n\|_{H^1_0}^2 - \tfrac{\varepsilon_n}{\theta}\|u_n\|_{H^1_0} - \tfrac{1 + \theta}{\theta} M_0 \|f_{M_0}\|_{L^1}\quad \text{for every }n\in \mathbb{N}.\] Now, supposing that \(\|u_{n_k}\|_{H^1_0} \to \infty\) for some increasing sequence \((n_k)\), we get \(I(u_{n_k}) \to \infty\) as well, but this clearly contradicts the assumption that \((u_n)\) is the Palais-Smale sequence for \(I\). Hence \((u_n)\) is bounded. ◻
With the above lemma at hand, we turn to the next result concerning the existence of a nontrivial solution to problem 34 . The proof is based on abstract results which implicitly rely on the direct method of the calculus of variations and on the Mountain Pass Theorem.
Theorem 5. Assume that 20 and 23 hold. If additionally one of the following condition holds:
then problem 34 has at least one nonzero solution.
Proof. We split the proof into two cases:
If assumptions 27 and 28 are satisfied, then Theorem 4 guarantees the existence of a solution \(u^*\) satisfying condition 36 . Note that \(I(0)=0\), and let \(u\in H^1_0(V)\setminus\{0\}\) be arbitrary. Setting \(c=\|u\|_{H^1_0}^2\) in Lemma 7 and taking any \(t\in(0,t^*)\), we obtain \[\label{equ:proof:thm:nonzero95solution} I(tu)=\psi(tu)+\Phi(tu)<\tfrac{t^2}{2}\|u\|_{H^1_0}^2 - t^2\|u\|_{H^1_0}^2 = -\tfrac{t^2}{2}\|u\|_{H^1_0}^2 < 0.\tag{39}\] Consequently, \(I(0) > I(u^*)\) and hence \(u^* \neq 0\).
If, on the other hand, assumptions 33 and 29 hold, then the conclusion follows directly from Proposition 2. Indeed, by combining Lemmas 1, 5, 9 and 10, and Proposition 3, we get that the functional \(I\) satisfies the Palais–Smale condition. It therefore remains to verify assumption ?? . To this end, set \(r=\frac{R}{9}\). Then, by Lemma 9, we obtain \[I(u)=\tfrac{1}{2}\mathcal{E}_a(u)-\alpha \geq \tfrac{a_-}{2}\|u\|_{H^1_0}^2-\alpha = \tfrac{a_-R^2}{162}-\alpha > 0 = I(0) \quad \text{for every } u\in \partial B_r.\] The second condition in ?? follows from Lemma 8. Indeed, let \(u\in H^1_0(V)\setminus\{0\}\) and \(t>0\). Using the estimate obtained there, we compute \[I(tu)=\tfrac{1}{2}\mathcal{E}_a(tu)-\int_V F\big(\boldsymbol{x},tu(\boldsymbol{x})\big)\,d\mu(\boldsymbol{x}) \leq \tfrac{t^2}{2}\mathcal{E}_a(u)-A t^\theta \|u\|_{L^\theta}^\theta-\|B\|_{L^1}.\] Letting \(t\to\infty\), we obtain \(I(tu)\to -\infty\). Hence, by choosing \(e=tu\) for \(t\) sufficiently large, condition ?? is satisfied. Consequently, Proposition 2 applies and yields the desired conclusion.
◻
By combining the proofs of Theorems 4 and 5, and by applying Proposition 2, we can prove the following result.
Theorem 6. Assume that 20 , 23 , 28 , 33 and 29 hold. Then problem 34 has at least two nonzero solutions.
Proof. In the second part of the proof of Theorem 5 we have already shown that, under the assumptions of the present theorem, the functional \(I\) satisfies the Palais–Smale condition and that ?? holds. In order to apply Proposition 2, it therefore remains to verify ?? . This is, however, an immediate consequence of Lemma 7. Indeed, fix an arbitrary nonzero element \(u \in H^1_0(V)\). By Lemma 7, repeating the computation in 39 , we obtain that there exists \(t^* > 0\) such that \(I(tu) < 0\) for every \(t \in (0,t^*)\). Hence, by Proposition 2, the problem 34 admits at least two nontrivial solutions. ◻
The authors declare that no funds, grants, or other support were received during the preparation of this manuscript.
This paper has been completed while the second author was a Doctoral Candidate in the Interdisciplinary Doctoral School at the Lodz University of Technology, Poland. The author’s contribution to this research was estimated at 50%.