April 16, 2026
We consider the following class of mixed local-nonlocal equations: \[\begin{align} \label{abs}} -\Delta_p u + (-\Delta)_p^s u = V \abs{u}^{p-2}u \text{ in } \Omega, \end{align}\] {#eq:abs} where \(s \in (0,1), p \in (1, \infty)\), and the weight function \(V\) lies in scaling subcritical Lebesgue space \(L^q(\Omega)\) where \(q>\frac{d}{p}\) when \(d>p\) and \(q>1\) when \(d \le p\). We establish Harnack inequality for weak solution and weak Harnack inequality for weak supersolution to eq:abs? . Our approach is based on the De Giorgi-Nash-Moser theory, the expansion of positivity and estimates involving a tail term. Our results also apply to integro-differential operators, with the prototype given by \((-\Delta)_p^s\). This work generalizes some regularity results of Garain-Kinnunen (Trans. Am. Math. Soc., 375(8), 2022) and Garain (Nonlinear Anal., 256, 2025) to the setting of general weight functions.
This paper studies Harnack inequality for a jump-diffusion process, which is described by the following homogeneous equation: \[\begin{align} \label{main95PDE}} -\Delta_p u + (-\Delta_p)^s u = V \abs{u}^{p-2}u \text{ in } \Omega, \end{align}\tag{1}\] where \(p \in (1, \infty)\) is the integrability exponent, \(s\in (0,1)\) is the differentiability parameter, \(\Omega\) is a bounded open set in \(\mathbb{R}^d\) and the weight function \(V\) satisfies \[\label{weight} V \in L^q(\Omega) \text{ with } \left\{\begin{align} &q>\frac{d}{p},\,&\text{if}\;d>p; \\ &q> 1,\,&\text{if}\;d\le p. \end{align} \right.\tag{2}\] The \(p\)-Laplace operator \(\Delta_p\) and the fractional \(p\)-Laplace operator \((-\Delta)_p^s\) are defined as \[\begin{align} \Delta_p u = \text{div}(\abs{ \nabla u}^{p-2} \nabla u), \text{ and } (-\Delta_p)^s u = \text{P.V.} \int _{\mathbb{R}^d} \frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))}{|x-y|^{d+ps}} {\,\rm d}y, \, x \in \mathbb{R}^d, \end{align}\] respectively, where P.V. means “in the principal value sense”.
The classical Harnack inequality, formulated in 1887, asserts that for any nonnegative harmonic function \(u: B_1 \rightarrow {\mathbb{R}}\), there exists a constant \(C>0\) such that the inequality \(u(x) \leq C u(y)\) holds for every \(x, y \in B_{1/2}\) (where \(B_r \subset \mathbb{R}^d\) is a ball of radius \(r\) with centre at origin). This inequality is known as the Harnack inequality, and it received significant interest after Moser in [1] demonstrated that Harnack inequality leads to a priori estimates in Hölder spaces. Subsequently, De-Giorgi (1956), Nash (1958), and Moser (1964) independently established the Harnack inequality for the weak solutions to \(- \text{div} (A(x) \nabla u) = 0\) in \(B_1\), where \(A\) is a bounded, measurable, and positive definite function. More precisely, for a nonnegative weak solution \(u\), there exists a constant \(C>0\) such that \[\begin{align} \sup_{B_{\frac{1}{2}}} u \le C \inf_{B_{\frac{1}{2}}} u. \end{align}\] This result has significant value in solving Hilbert’s 19th Problem. Since then, Harnack inequalities for weak solutions of various local elliptic operators have been extensively studied. We refer to [2]–[4] for detailed descriptions of this study. In [5], Chiarenza-Fabes-Garofalo established the Harnack inequality for any local positive weak solution to \(-\Delta u = fu \; \text{in} \; \Omega,\) where the function \(f\) lies in a scaling critical class, namely in the Stummel class of potentials. Later, in [6], Biroli extended this result to a nonlinear set-up. It is shown that if \(u\) is a local positive weak solution to \(-\Delta_p u = f |u|^{p-2}u\) in \(\Omega\), where \(f\) lies in the Kato space, then there exists \(C>0\) such that \[\begin{align} \sup_{B_r(x)} u \le C \inf_{B_r(x)} u, \end{align}\] for every \(B_r(x) \subsetneq \Omega\). Their proof mainly followed Moser’s approach with the application of classical John-Nirenberg Lemma (see [7] and [3]).
We now highlight the contribution of Castro-Kuusi-Palatucci [8], who initiated the study of Harnack inequalities for a quasilinear nonlocal operator. They introduced the tail term, denoted as \(\text{\rm Tail}_{p-1,sp,p}\) (see 9 ) in proving the Harnack estimates for the following problem \[\begin{align} \label{Pala} (-\Delta_p)^s u = 0 \text{ in } \Omega, \; u=g \text{ in } \mathbb{R}^d\setminus \Omega, \end{align}\tag{3}\] where \(\Omega\) is a bounded open set in \(\mathbb{R}^d\). More precisely, they have shown that if \(u\) is a local nonnegative weak solution to 3 , then there exists a constant \(C=C(d,p,s)>0\) such that \[\begin{align} \label{nonlocal-harnack} \sup_{B_r(x)} u \le C \inf_{B_r(x)} u + C\left( \frac{r}{R}\right)^{\frac{sp}{p-1}} \text{\rm Tail}_{p-1,sp,p}(u^-,x; R), \end{align}\tag{4}\] for every \(B_r(x) \subset B_{R/2}(x)\) with \(x \in \Omega\) and \(B_R(x) \subset \Omega\). In particular, if \(u\) is nonnegative in \(\mathbb{R}^d\), then 4 reduces to the classical Harnack inequality. A counterexample due to Kassmann [9] shows that such positivity assumptions cannot be removed or weakened to get classical Harnack inequality, even in the case \(p=2\), i.e., for the fractional Laplacian \((-\Delta)^s\). In [8], the authors also studied a weak Harnack type inequality for nonnegative weak supersolutions to 3 . It is worth noting that the authors have employed De Giorgi’s approach to establish these regularity results. We also mention the works of [10]–[14], where significant developments concerning interior Hölder regularity, higher Hölder regularity, Lipschitz regularity, and \(\mathcal{C}^{1,\alpha}\) regularity for weak solutions to \((-\Delta_p)^s u =f\) in \(\Omega\), (for a suitable potential function \(f\)) have been investigated.
To provide proper context for our results, we now review the literature associated with the regularity of mixed local-nonlocal problems. We begin with mentioning the work of Garain-Kinnunen [15], where, following the ideas developed in [8], the authors have obtained the local boundedness, Harnack inequality, weak Harnack inequality and local Hölder regularity for weak solutions to \[\label{mixed} - \Delta_p u+(-\Delta_p)^s u=0 \text{ in } \Omega,\tag{5}\] where \(\Omega\) is a bounded open set in \(\mathbb{R}^d\). The higher Hölder regularity and almost Lipschitz regularity for weak solutions to 5 are obtained by Garain-Lindgren [16]. In [17], Filippis-Mingione studied the following mixed local-nonlocal problem with a potential \(f\): \[\label{MinFi} - \Delta_p u+(-\Delta_r)^s u=f \text{ in } \Omega,\tag{6}\] where \(p,r\in (1, \infty)\) with \(p\geq sr\) and \(\Omega\) is a bounded open set in \(\mathbb{R}^d\). It is shown that weak solutions to 6 are locally Hölder continuous for every \(\alpha\in(0,1)\) when \(f \in L^d(\Omega)\), and their gradients are locally Hölder continuous for some exponent \(\alpha \in(0,1)\) when \(f \in L^q(\Omega)\) with \(q>d\). In addition, they have obtained boundary regularity of weak solutions to 6 . We refer [17] for more details on the technical assumptions related to their operators. The work of [17] has been recently complemented by Biswas-Topp in [18], who obtained interior \(\mathcal{C}^{1, \alpha}\)-regularity for weak solutions to 6 with \(p\leq sr\) and locally bounded \(f\). Finally, in [19], Antonini-Cozzi establish \(\mathcal{C}^{1, \alpha}\)-regularity for weak solutions, up to the boundary, with \(f \in L^q(\Omega)\) with \(q>d\). A general Hopf lemma is also obtained in [19]. For further interior and boundary regularity estimates for weak solutions to the mixed local-nonlocal operators, we refer to [20]–[26] and the references therein. In [27], Garain recently established the Harnack inequality for weak solutions and the weak Harnack inequality for weak supersolutions to 6 in the case \(r=p\), assuming \(f\in L^{q/p}(\Omega)\) for some \(q>d\). Two alternative proofs were provided: one based on the classical John-Nirenberg lemma (Moser’s approach), and the other on the Bombieri-Giusti lemma, combining inverse estimates and reverse Hölder inequalities for weak supersolutions, logarithmic estimates, and appropriate tail estimates. Finally, we mention the work of [28], where, using De Giorgi’s approach (as in [8]), the same author obtained Harnack and weak Harnack inequalities for weak solutions to 1 with \(V \in L^{\infty}(\Omega)\).
In this paper, we prove the following regularity properties for weak solutions, weak subsolutions, and weak supersolutions (Definition [maindef]) of 1 :
Local boundedness. In Lemma 5, we show that every weak subsolution to 1 is locally bounded. Our proof proceeds via an energy estimate (Lemma 2) combined with the Sobolev inequality and a Moser-type iteration scheme (Lemma 5). We emphasize that a direct adaptation of the arguments used in [8], [15] is not feasible in our case due to the presence of an unbounded weight function \(V\). By invoking the scaling properties detailed in Remark 4, we may assume without loss of generality that, \(u\) weakly solves the scaled equation \(-\Delta_p u + \theta(-\Delta_p)^s u = V\abs{u}^{p-2}u \text{ in } \Omega\), where \(\theta \in (0,1]\) and \(\@ifstar{\norm}{\norm*}{V}_{L^q(\Omega)}\) is sufficiently small. This smallness condition is essential for maintaining control over the constants within the estimates. Furthermore, the statement of local boundedness (see ?? ) introduces a parameter \(\sigma\), where the prescribed integrability of \(V\) ensures the strict inequalities \(\sigma>1\) and \(p\sigma<p^*\), which are vital for the convergence in the iteration process. Notably, if \(V\) lies in the scaling critical space \(L^{d/p}(\Omega)\), then \(p \sigma = p^*\) causes the iteration process to fail, thereby precluding the derivation of local boundedness via this framework.
Harnack inequality and weak Harnack inequality. In Theorem [harnack95intro] we derive the Harnack inequality for weak solutions to 1 , and in Theorem [weakhar95intro] we establish the weak Harnack inequality for weak super solutions to 1 . Our argument follows the strategy developed by Di Castro, Kuusi, and Palatucci [8]. In this approach, both the local boundedness and logarithmic energy estimate (Lemma 4) are fundamental components. Additionally, the expansion of positivity (Lemma [expan]) and a suitable tail estimate (Lemma 6) play essential roles in the proof. The exponent \(\sigma\) is again a key ingredient in establishing the expansion of positivity. The smallness assumption on \(\|V\|\) and the strict inequality \(p\sigma <p^*\) are also required at this stage.
Our results are applicable to a broader class of nonlocal operators defined by \[\mathcal{L} u(x)=\text{P.V.} \int_{\mathbb{R}^d} K_{\text{sym }}(x, y)|u(x)-u(y)|^{p-2}(u(x)-u(y)) {\,\rm d}y, \quad x \in \mathbb{R}^d ;\] where \(K\) is a suitable kernel of order \((s, p)\) with merely measurable coefficients. The function \(K_{\text{sym }}\) is the symmetric part of \(K\) defined as \(K_{\text{sym }}(x, y)=(K(x, y)+K(y, x)) / 2\), where \(K: \mathbb{R}^d \times \mathbb{R}^d \rightarrow[0, \infty)\) is a measurable function such that \[\lambda \leq K(x, y)|x-y|^{d+sp} \leq \Lambda, \text{ for a.e. } x, y \in \mathbb{R}^d,\] where \(\lambda \geq \Lambda \geq 1\). Further, the assumption on \(K\) can be weakened as follows \[\begin{array}{ll} \lambda \leq K(x, y)|x-y|^{d+s p} \leq \Lambda & \text{ for a.e. } x, y \in \mathbb{R}^d \text{ s.t. }|x-y| \leq 1, \\ 0 \leq K(x, y)|x-y|^{d+\eta} \leq M & \text{ for a.e. } x, y \in \mathbb{R}^d \text{ s.t. }|x-y|>1, \end{array}\] for some \(\lambda, \Lambda\) as above, \(\eta>0\) and \(M \geq 1\).
Remark 1. We do not address local Hölder continuity in this work, since it follows from [16] (at least in the case \(p\geq 2\)) with the fact that every weak solution to 1 is in \(L^{\infty}(\Omega)\).
The rest of the paper is organized as follows. In Section [prelims], we introduce the relevant function spaces and present the main results. Section [energylog] is devoted to establishing energy and logarithmic estimates for weak solutions to the scaled equation 11 . In Section [localbdd], we investigate the local boundedness of these weak solutions within our framework. Finally, Section [harnackinq] provides the proofs of the expansion of positivity, the Harnack and weak Harnack inequalities.
For an open set \(E \subset \mathbb{R}^d\), the Sobolev space \(W^{1,p}(E)\) and the fractional Sobolev space \(W^{s,p}(E)\) are defined as \[\begin{align} &W^{1,p}(E) := \left\{ u \in L^p(E) : \int_{E} \abs{ \nabla u(x)}^p {\,\rm d}x< \infty \right\}, \\ &W^{s,p}(E):= \left\{ u \in L^p(E) : \iint_{E \times E} \frac{|u(x)-u(y)|^p}{|x-y|^{d+sp}} {\,\rm d}x{\,\rm d}y< \infty \right\}, \end{align}\] which are endowed with the following norms respectively: \[\begin{align} &\@ifstar{\norm}{\norm*}{u}_{W^{1,p}(E)}:= \left( \int_{E} \abs{u(x)}^p {\,\rm d}x+ \int_{E} \abs{\nabla u(x)}^p {\,\rm d}x\right)^{\frac{1}{p}}, \\ &\@ifstar{\norm}{\norm*}{u}_{W^{s,p}(E)} := \left( \int_{E} \abs{u(x)}^p {\,\rm d}x+ \iint_{E \times E} \frac{|u(x)-u(y)|^p}{|x-y|^{d+sp}} {\,\rm d}x{\,\rm d}y\right)^{\frac{1}{p}}. \end{align}\] Then we consider the Sobolev space \(W_0^{1,p}(E)\), defined as \(W_0^{1,p}(E) := \{ u \in W^{1,p}(\mathbb{R}^d) : u=0 \text{ in } \mathbb{R}^d\setminus E\}\). From [29], the following continuous embedding holds: \[\begin{align} \@ifstar{\norm}{\norm*}{u}_{W^{s,p}(E)} \le C(d,p,s) \@ifstar{\norm}{\norm*}{u}_{W^{1,p}(E)}, \; \forall \, u \in W^{1,p}(E). \end{align}\]
Now we recall the following Gagliardo-Nirenberg-Sobolev inequality, see [2].
Lemma 1. Let \(1<p<\infty\) and \(E\) be an open set in \({\mathbb{R}}^d\) with \(\lvert E\rvert<\infty\) and \[\label{kappa} \kappa= \begin{cases} \frac{d}{d-p},&\text{if}\quad d>p,\\ 2,&\text{if}\quad d \leq p. \end{cases}\tag{7}\] Then there exists a positive constant \(C=C(d,p)\) such that \[\label{e46friedrich} \biggl(\int_E \lvert u(x)\rvert^{\kappa p} {\,\rm d}x\biggr)^{\frac{1}{\kappa p}} \le C(d,p) \lvert E \rvert^{\frac{1}{d}-\frac{1}{p}+\frac{1}{\kappa p}} \@ifstar{\norm}{\norm*}{u}_{W^{1,p}(E)},\tag{8}\] for every \(u\in W^{1,p}(E)\).
In the study of nonlocal equations, the global behaviour of solutions comes into play. This is entailed by the tail space \[L^{q}_{\alpha}(\mathbb{R}^d)=\left\{u\in L^{q}_{\rm loc}(\mathbb{R}^d):\int_{\mathbb{R}^d} \frac{|u|^q}{1+|x|^{d+\alpha}}\,{\,\rm d}x<+\infty\right\},\; q>0and\alpha>0,\] and measured by the quantity \[\label{mtail} \mathrm{Tail}_{q,\alpha,\beta}(u;x_0,R)=\left(R^{\beta}\,\int_{\mathbb{R}^d\setminus B_R(x_0)} \frac{|u|^q}{|x-x_0|^{d+\alpha}}{\,\rm d}x\right)^\frac{1}{q},\tag{9}\] defined for every \(x_0\in\mathbb{R}^d\), \(R>0,\,\beta>0\) and \(u\in L^q_{\alpha}(\mathbb{R}^d)\). We observe that 9 is always finite, for a function \(u\in L^q_{\alpha}(\mathbb{R}^d)\).
Next, we recall the definitions of supersolution, subsolution, and solution of 1 .
Definition 1. A function \(u\in W^{1,p}_{loc}(\Omega) \cap L_{sp}^{p-1}(\mathbb{R}^d)\) is a weak supersolution and subsolution to 1 if for every \(K \subset \subset \Omega\) and \(v\in W_0^{1,p}(K)\) with \(v\geq0\) a.e. in \(K\), it holds \[\begin{align} \label{weak1} \int_{\Omega}|\nabla u|^{p-2}\nabla u\cdot\nabla v \,{\,\rm d}x+ {\mathcal{A}}(u,v) \geq \text{ or } \leq \int_{\Omega} V|u|^{p-2}uv {\,\rm d}x. \end{align}\tag{10}\] We say \(u\) is a weak solution if the equality holds in 10 for every \(v\in W_0^{1,p}(K)\).
The main results of this paper are stated below.
Theorem 2 (Harnack Inequality). Let \(s \in (0,1), p \in (1, \infty)\), and \(V \in L^q(\Omega)\) for \(q\) as given in 2 . Let \(R >0\) and \(x_0 \in \Omega\) be such that \(B_{R}(x_0)\subset \Omega\), and let \(u\) be a weak solution of 1 satisfying \(u \ge 0\) in \(B_R(x_0)\). Then there exists \(R_0 < R\) such that for \(r \in (0, \min\{\frac{R_0}{2},1\}]\), the following holds \[\mathop{\mathrm{ess\,sup}}_{B_{\frac{r}{2}}(x_0)} u \le C \mathop{\mathrm{ess\,inf}}_{B_r(x_0)} u + C\left(\frac{r}{R_0}\right)^{\frac{p}{p-1}} \text{\rm Tail}_{p-1,sp,p}(u^-; x_0, R_0),\] where \(C=C(d,p,s)>0\) is a constant.
Theorem 3 (Weak Harnack Inequality). Let \(s \in (0,1), p \in (1, \infty)\), and \(V \in L^q(\Omega)\) for \(q\) as given in 2 . Let \(R >0\) and \(x_0 \in \Omega\) be such that \(B_{R}(x_0)\subset \Omega\), and let \(u\) be a weak supersolution of 1 satisfying \(u \ge 0\) in \(B_R(x_0)\). Then there exists \(R_0 < R\) such that for \(r \in (0, \min\{\frac{R_0}{2},1\}]\), the following holds \[\left(\fint_{B_{\frac{r}{2}}\left(x_0\right)} u^l {\,\rm d}x\right)^{\frac{1}{l}} \leq C \mathop{\mathrm{ess\,inf}}_{B_r(x_0)} u+C\left(\frac{r}{R_0}\right)^{\frac{p}{p-1}} \text{\rm Tail}_{p-1,sp,p}\left(u^{-} ; x_0, R_0\right),\] whenever \(0<l<\kappa(p-1)\), with \(\kappa\) as given in 7 . Here \(C=C(d,p,s)>0\) is a constant.
We require the following remark.
Remark 4. (a) Take \(\rho \in (0,1)\) and \(x_0 \in \Omega\) such that \(B_{\rho}(x_0) \subset \Omega\). By translation invariance of the operator, we may assume that \(x_0 =0\). Then for a weak solution \(u\) of 1 , the following holds weakly \[-\Delta_p u + (-\Delta)_p^s u = V \abs{u}^{p-2}u, \text{ in } B_{\rho}.\] Define \(u_\rho(x) := u(\rho x)\). Then \(u_\rho\) satisfies the following equation weakly \[\begin{align} -\Delta_p u_{\rho} + \rho^{p-sp} (-\Delta)_p^s u_\rho & = \rho^p \left( -\Delta_p u(\rho x) + (-\Delta)_p^s u(\rho x) \right) \\ &= \rho^p V(\rho x) \abs{u(\rho x)}^{p-2}u(\rho x) = V_\rho(x) \abs{u_\rho}^{p-2} u_\rho, \text{ in } B_1, \end{align}\] where \(V_{\rho}(x) = \rho^p V(\rho x)\). Moreover, for \(q\) as given in 2 , \[\begin{align} \@ifstar{\norm}{\norm*}{V_\rho}_{L^{q}(B_1)} = \rho^{p-\frac{d}{q}} \@ifstar{\norm}{\norm*}{V}_{L^{q}(B_\rho)} = o(\rho), \end{align}\] as \(\rho \rightarrow 0\).
(b) Now we consider \(\tilde{\Omega} := B_1 \cap \Omega\). Since \(0 \in \Omega\), we observe that \(|\tilde{\Omega}| >0\). From (a), \(u_{\rho}\) weakly solves the following equation \[\begin{align} -\Delta_p u_{\rho} + \rho^{p-sp} (-\Delta)_p^s u_\rho = V_\rho(x) \abs{u_\rho}^{p-2} u_\rho, \text{ in } \tilde{\Omega}, \end{align}\] where \(\@ifstar{\norm}{\norm*}{V_\rho}_{L^q(\tilde{\Omega})}\) is sufficiently small depending on a fixed postive value of \(\rho\).
In view of the above remark, from now onward, we consider the following equation \[\begin{align} \label{eqn-scaled}_{\theta}} -\Delta_p u + \theta (-\Delta)_p^s u = V \abs{u}^{p-2}u, \text{ in } \Omega; \quad \theta \in (0,1], \end{align}\tag{11}\] with sufficiently small \(\|V\|_{L^q(\Omega)}\) where \(q\) as given in 2 . We aim to establish the Harnack and weak Harnack estimates for weak solutions and weak super solutions to 11 , as presented in theorems [harnack] and [weakharnack], respectively.
Notation and Convention. We fix the following notations and conventions to be used in this paper:
(a) We denote \[\begin{align} &\,{\rm d}\mu:= |x-y|^{-(d+sp)} {\,\rm d}x{\,\rm d}y, \; A_u(x,y):= |u(x)-u(y)|^{p-2}(u(x)-u(y)), \text{ and } \\ &\mathcal{A}(u,v):= \iint_{\mathbb{R}^d\times \mathbb{R}^d}A_u(x,y)(v(x)-v(y)) \,{\rm d}\mu. \end{align}\]
(b) For \(d>p\), \(p^*=\frac{d p}{d-p}\) is the critical Sobolev exponent.
(c) \(u^\pm := \max\{\pm u, 0\}\) denote the positive and negative parts of \(u\).
(d) We denote \(\text{\rm Tail}_{p-1,sp,p}(\cdot ;\cdot, \cdot)\) by \(\text{\rm Tail}(\cdot ;\cdot, \cdot)\).
(e) We always take \(x_0 \in \Omega\) and \(r>0\) such that \(B_r(x_0) \subset \Omega\). For brevity, we denote \(B_r(x_0)\) as \(B_r\).
(f) \(C,C_i\) (where \(i=1,2, \cdots\)) denote generic positive constants.
(g) Throughout the paper, we consider the case \(d>p\). For \(d\le p\), proof follows using similar set of arguments.
We begin with the following energy estimate for weak solutions to 11 .
Lemma 2. Let \(V \in L_{loc}^1(\Omega)\), and \(u\) be a weak subsolution to 11 . Let \(w=(u-k)^+\) for \(k \in {\mathbb{R}}\). Then there exists \(C=C(p)\) such that \[\begin{align} {\label{cacc}} &\int_{B_{r}} \phi^{p} |\nabla w|^{p} {\,\rm d}x+ \theta \iint_{B_{r} \times B_{r}} |w(x)\phi(x) - w(y)\phi(y)|^{p} \,{\rm d}\mu\nonumber\\ & \le C \Bigg( \int_{B_{r}} w^{p} |\nabla \phi|^{p} {\,\rm d}x+ \int_{B_{r}}| V(x)| |u|^{p-1} w \phi^p {\,\rm d}x+ \theta \iint_{B_{r} \times B_{r}} \max\{w(x), w(y)\}^{p} |\phi(x)-\phi(y)|^{p} \,{\rm d}\mu\nonumber \\ &+ \theta \mathop{\mathrm{ess\,sup}}_{x \in \text{supp} (\phi)} \int_{\mathbb{R}^d\setminus B_{r}} \frac{w(y)^{p-1}}{|x-y|^{d+ps}} {\,\rm d}y\cdot \int_{B_{r}} w\phi^{p} {\,\rm d}x\Bigg), \end{align}\tag{12}\] where \(\phi \in {\mathcal{C}}_c^{\infty}(B_r)\) is a nonnegative function. If \(u\) is a weak supersolution of 11 , the estimate in 12 holds with \(w=(u-k)^-\).
Proof. Since \(u\) is a subsolution to 11 , taking \(v:= w \phi^{p} \in W_{0}^{1,p}(\Omega)\) as a nonnegative test function, we write \[\int_{\Omega} |\nabla u|^{p-2} \nabla u \cdot \nabla v {\,\rm d}x+ \theta {\mathcal{A}}(u, v) \le \int_{\Omega} V|u|^{p-2}uv {\,\rm d}x.\] Using the above inequality, we obtain \[\begin{align} \label{CC1} \int_{\Omega} |\nabla w|^{p} \phi^{p} {\,\rm d}x+ \theta{\mathcal{A}}(u, v) & = \int_{\Omega} |\nabla u|^{p-2} \nabla u \cdot \nabla v {\,\rm d}x+ \theta{\mathcal{A}}(u,v) - p \int_{\Omega} w \phi^{p-1} |\nabla w|^{p-2} \nabla w \cdot \nabla \phi {\,\rm d}x\nonumber \\ & \le \int_{\Omega} |V||u|^{p-1}v {\,\rm d}x- p \int_{\Omega} w \phi^{p-1} |\nabla w|^{p-2} \nabla w \cdot \nabla \phi {\,\rm d}x. \end{align}\tag{13}\] From the nonlocal energy estimate [30], we have \[\begin{align} \label{CC2} {\mathcal{A}}(u, v) & \ge C(p) \iint_{B_{r} \times B_{r}} |w(x)\phi(x) - w(y)\phi(y)|^{p} \,{\rm d}\mu- C(p) \iint_{B_{r} \times B_{r}} \max\{w(x), w(y)\}^{p} |\phi(x)-\phi(y)|^{p} \,{\rm d}\mu\nonumber \\ & - C(p) \mathop{\mathrm{ess\,sup}}_{x \in \text{supp}(\phi)} \int_{\mathbb{R}^d\setminus B_{r}} \frac{w(y)^{p-1}}{|x-y|^{d+ps}} {\,\rm d}y\cdot \int_{B_{r}} w\phi^{p} {\,\rm d}x. \end{align}\tag{14}\] Further, using Young’s inequality, for \(\varepsilon>0\), \[\begin{align} p \left| \int_{\Omega} w \phi^{p-1} |\nabla w|^{p-2} \nabla w \cdot \nabla \phi {\,\rm d}x\right| & \le p \int_{\Omega} |w| \phi^{p-1} |\nabla w|^{p-1} |\nabla \phi| {\,\rm d}x\nonumber \\ & \le \varepsilon \int_{B_{r}} |\nabla w|^{p} \phi^{p} {\,\rm d}x+ O\left(\frac{1}{\varepsilon}\right) C(p) \int_{B_{r}} |\nabla \phi|^{p} w^{p} {\,\rm d}x. \end{align}\] By taking \(\varepsilon = (2(2^{p}+1))^{-1}\), and combining 13 and 14 , the required estimate holds. In the case of a weak supersolution, the estimate in 12 follows by applying the obtained result to \(-u\). ◻
The following lemma establishes an energy estimate for a weak supersolution to 11 , which will be helpful in proving the weak Harnack inequality.
Lemma 3. Assume that \(u\) is a weak supersolution to 11 satisfying \(u \geq 0\) in \(B_R(x_0) \subset \Omega\). Let \(B_r (x_0)\subset B_{\frac{3R}{4}}(x_0)\). Denote \(w=(u+t)^{\frac{p-m}{p}}\) where \(m \in(1, p)\) and \(t>0\). Then there exists \(\zeta_1(m,d,p)>0\) such that if \(\|V\|_{L^q(\Omega)} \le \zeta_1\), then \[\begin{align} \int_{B_r} \phi^p|\nabla w|^p {\,\rm d}x\leq C(p) & \left(\frac{(p-m)^p}{(m-1)^{\frac{p}{p-1}}} \int_{B_r} w^p|\nabla \phi|^p {\,\rm d}x\right. \\ & +\theta\frac{(p-m)^p}{(m-1)^p} \iint_{B_r \times B_r} \max \{w(x), w(y)\}^p|\phi(x)-\phi(y)|^p \,{\rm d}\mu\\ & +\theta\frac{(p-m)^p}{(m-1)}\bigg(\underset{z \in \operatorname{supp} \phi}{\operatorname{ess} \sup } \int_{\mathbb{R}^d \backslash B_r} \frac{{\,\rm d}y}{|z-y|^{d+ps}} \\ & \left.+t^{1-p} R^{-p} \operatorname{Tail}\left(u^{-} ; x_0, R\right)^{p-1}\bigg) \int_{B_r} w^p \phi^p {\,\rm d}x\right), \end{align}\] for every \(\phi \in {\mathcal{C}}_c^{\infty}(B_r)\) with \(\phi \ge 0\).
Proof. Let \(t>0, h=u+t\) and \(m \in[1+\varepsilon, p-\varepsilon]\) for \(\varepsilon>0\) small enough. As \(u\) is a weak supersolution of 11 , by choosing \(\phi=h^{1-m} \phi^p\) as a test function in 11 , we obtain \[\begin{align} {\label{1}} & 0 \leq\int_{B_r}|\nabla u|^{p-2} \nabla u \cdot \nabla(h^{1-m} \phi^p) {\,\rm d}x\nonumber\\ &+\theta\iint_{B_r \times B_r} |u(x)-u(y)|^{p-2}(u(x)-u(y))(h(x)^{1-m} \phi(x)^p-h(y)^{1-m} \phi(y)^p) \,{\rm d}\mu\nonumber\\ &+2 \theta\iint_{\mathbb{R}^d \backslash B_r \times B_r} |u(x)-u(y)|^{p-2}(u(x)-u(y))h(x)^{1-m} \phi(x)^p \,{\rm d}\mu+\int_{B_{r}} |V| \frac{u^{p-1} \phi^p}{(u+t)^{m-1}} {\,\rm d}x\nonumber\\&= \int_{B_r}|\nabla h|^{p-2} \nabla h \cdot \nabla(h^{1-m} \phi^p) {\,\rm d}x\nonumber\\ &+\theta\iint_{B_r \times B_r} |h(x)-h(y)|^{p-2}(h(x)-h(y))(h(x)^{1-m} \phi(x)^p-h(y)^{1-m} \phi(y)^p) \,{\rm d}\mu\nonumber\\ &+2\theta \iint_{\mathbb{R}^d \backslash B_r \times B_r} |h(x)-h(y)|^{p-2}(h(x)-h(y))h(x)^{1-m} \phi(x)^p \,{\rm d}\mu+\int_{B_{r}} |V| \frac{u^{p-1} \phi^p}{(u+t)^{m-1}} {\,\rm d}x\nonumber\\ &=:I_1+ \theta I_2+2\theta I_3+I_4. \end{align}\tag{15}\]
Estimate of \(I_1\). We observe that \[\begin{align} {\label{I1}} I_1 & =\int_{B_r}|\nabla h|^{p-2} \nabla h \cdot \nabla\left(h^{1-m} \phi^p\right) {\,\rm d}x\nonumber\\ & \leq(1-m) \int_{B_r} h^{-m}|\nabla h|^p \phi^p {\,\rm d}x+p \int_{B_r} h^{1-m}|\nabla \phi||\nabla h|^{p-1} \phi^{p-1} {\,\rm d}x\nonumber\\ & =(1-m) J_1+J_2, \end{align}\tag{16}\] where \[J_1=\int_{B_r} h^{-m}|\nabla h|^p \phi^p {\,\rm d}x,\; \text{ and }\; J_2=p \int_{B_r} h^{1-m}|\nabla \phi||\nabla h|^{p-1} \phi^{p-1} {\,\rm d}x.\] Now, by Young’s inequality, we obtain \[\begin{align} {\label{J2}} J_2 & \leq \frac{m-1}{2} J_1+\frac{C(p)}{(m-1)^{\frac{1}{p-1}}} \int_{B_r}|\nabla \phi|^p h^{p-m} {\,\rm d}x. \end{align}\tag{17}\] By applying 17 in 16 , for some constant \(C=C(p)>0\), we have \[\begin{align} {\label{I11}} I_1 & \leq \frac{1-m}{2} \int_{B_r} h^{-m}|\nabla h|^p \phi^p {\,\rm d}x+\frac{C}{(m-1)^{\frac{1}{p-1}}} \int_{B_r}|\nabla \phi|^p h^{p-m} {\,\rm d}x\nonumber\\ & =-\frac{m-1}{2}\left(\frac{p}{p-m}\right)^p \int_{B_r}\left|\nabla\left(h^{\frac{p-m}{p}}\right)\right|^p \phi^p {\,\rm d}x+\frac{C}{(m-1)^{\frac{1}{p-1}}} \int_{B_r}|\nabla \phi|^p h^{p-m} {\,\rm d}x. \end{align}\tag{18}\]
Estimates of \(I_2\) and \(I_3\). Following the lines of the proof of [8] for \(w=h^{\frac{p-m}{p}}\), with some positive constants \(C(p,m)\) and \(C(p)\), we obtain \[\begin{align} {\label{I2}} & I_2+I_3 \leq -C(p,m) \iint_{B_r \times B_r}|w(x)-w(y)|^p \phi(y)^p \,{\rm d}\mu\nonumber\\ & +\frac{C(p)}{(m-1)^{p-1}} \iint_{B_r \times B_r} \max \{w(x), w(y)\}^p|\phi(x)-\phi(y)|^p \,{\rm d}\mu\nonumber\\&+C\left(\mathop{\mathrm{ess\,sup}}_{z \in \operatorname{supp} \phi} \int_{\mathbb{R}^d \backslash B_r} \frac{1}{|z-y|^{d+ps}}{\,\rm d}y+t^{1-p} \int_{\mathbb{R}^d \backslash B_r}\frac{((u(y))^{-})^{p-1}}{|y-x_0|^{d+ps}} {\,\rm d}y\right) \int_{B_r} w^p \phi^p {\,\rm d}x. \end{align}\tag{19}\]
Estimate of \(I_4\). Noting \(t>0\), and the fact that \(\phi w=\phi(u+t)^{\frac{p-m}{p}}\in W^{1,p}_0(\Omega)\), by Sobolev and Hölder inequality, it holds \[\begin{align} {\label{I4}} \int_{B_{r}} & |V| \frac{u^{p-1} \phi^p}{(u+t)^{m-1}} {\,\rm d}x\leq \int_{B_{r}} |V| u^{p-m} \phi^p{\,\rm d}x=\int_{B_{r}} |V| (\phi w)^p {\,\rm d}x\nonumber\\ &\leq C(d,p) \|V\|_{L^{q}(B_r)}|B_{r}|^{\frac{p}{d}-\frac{1}{q}} \left\| \phi w \right\|_{L^{p^*}(B_{r})}^p \nonumber\\ &\leq C(d,p) \|V\|_{L^{q}(B_r)} \int_{B_r} |\nabla(\phi w)|^p{\,\rm d}x\nonumber\\ &\leq C(d,p) \|V\|_{L^{q}(B_r)} \int_{B_r} |\nabla w|^p\phi ^p{\,\rm d}x+C(d,p) \|V\|_{L^{q}(B_r)} \int_{B_r} w^p|\nabla\phi |^p{\,\rm d}x. \end{align}\tag{20}\] We now choose \(\zeta_1=\zeta_1(d,p,m)>0\) in 20 so small such that \[C(d,p) \zeta_1 \leq \min\left\{\frac{m-1}{4}\left(\frac{p}{p-m}\right)^p, \frac{1}{(m-1)^{\frac{1}{p-1}}}\right\}.\] Thus, whenever \(\|V\|_{L^q(\Omega)}\leq \zeta_1\), \[\begin{align} {\label{I44}} I_4 & \leq\frac{m-1}{4}\left(\frac{p}{p-m}\right)^p \int_{B_r}\left|\nabla\left(h^{\frac{p-m}{p}}\right)\right|^p \phi^p {\,\rm d}x+\frac{C}{(m-1)^{\frac{1}{p-1}}} \int_{B_r}|\nabla \phi|^p h^{p-m} {\,\rm d}x. \end{align}\tag{21}\] By applying 18 , 19 and 21 in 15 , we conclude the proof. ◻
In the following lemma, we obtain a logarithmic energy estimate for weak supersolution.
Lemma 4. Assume that \(u\) is a weak supersolution to 11 satisfying \(u \geq 0\) in \(B_R(x_0) \subset \Omega\). Let \(B_r (x_0)\subset B_{\frac{R}{2}}(x_0)\) and \(t>0\). Then there exists \(C=C(d, p, s)\) such that the following holds: \[\begin{align} &\int_{B_r}|\nabla \log (u+t)|^p {\,\rm d}x+\theta\iint_{B_r \times B_r}\left|\log \left(\frac{u(x)+t}{u(y)+t}\right)\right|^p \,{\rm d}\mu\nonumber\\&\leq C r^d\left(r^{-p}+\theta r^{-p s}+\theta t^{1-p} R^{-p} \operatorname{Tail}\left(u^{-} ; x_0, R\right)^{p-1}\right). \end{align}\]
Proof. Consider a cut-off function \(\phi \in {\mathcal{C}}_c^{\infty}(B_{\frac{3 r}{2}})\) satisfying \(\phi = 1\) on \(B_r, 0 \leq \phi \leq 1\) and \(|\nabla \phi| \leq \frac{C}{r}\) on \(B_{\frac{3 r}{2}}\). By choosing \(v=(u+t)^{1-p} \phi^p \in W_0^{1, p}(\Omega)\) as a nonnegative test function we obtain \[\begin{align} {\label{log}} 0 &\leq \int_{B_{2 r}}|\nabla u|^{p-2} \nabla u \cdot \nabla\left(\frac{\phi^p}{(u+t)^{p-1}}\right) \mathrm{d} x+\theta\iint_{\mathbb{R}^d \times \mathbb{R}^d}|u(x)-u(y)|^{p-2}(u(x)-u(y)) \nonumber\\ &\quad\times\left(\frac{\phi(x)^p}{(u(x)+t)^{p-1}}-\frac{\phi(y)^p}{(u(y)+t)^{p-1}}\right) \mathrm{d} \mu+\int_{B_{2 r}} |V| \frac{u^{p-1} \phi^p}{(u+t)^{p-1}} {\,\rm d}x=:I_1+\theta I_s+I_2. \end{align}\tag{22}\] Let \(\varepsilon \in(0,1)\) be given. Applying the Young’s inequality, we estimate \(I_1\) as follows: \[\begin{align} {\label{I951}} I_1 & =\int_{B_{2 r}}|\nabla u|^{p-2} \nabla u \cdot\left((1-p) \frac{\phi^p \nabla u}{(u+t)^p}+p \frac{\phi^{p-1} \nabla \phi}{(u+t)^{p-1}}\right) \mathrm{d} x \nonumber\\ & \leq(1-p+\varepsilon) \int_{B_{2 r}} \frac{|\nabla u|^p \phi^p}{(u+t)^p} {\,\rm d}x+O\left(\frac{1}{\varepsilon}\right) C(p) \int_{B_{\frac{3 r}{2}}}|\nabla \phi|^p {\,\rm d}x. \end{align}\tag{23}\] By noticing the following that \[\begin{align} |\nabla \log (u+t)|^p=\frac{|\nabla u|^p}{(u+t)^p}, \text{ and } \int_{B_{\frac{3 r}{2}}}|\nabla \phi|^p {\,\rm d}x=C(d) \int_r^{\frac{3 r}{2}} \tau^{d-1-p} \mathrm{~d} \tau=C(d, p) r^{d-p} \end{align}\] and using 23 , we obtain \[\begin{align} {\label{log1}} I_1 \leq(1-p+\varepsilon) \int_{B_{2 r}}|\nabla \log (u+t)|^p \phi^p {\,\rm d}x+O\left(\frac{1}{\varepsilon}\right) C(d, p) r^{d-p}. \end{align}\tag{24}\] Now we estimate \(I_2\) as follows: \[\begin{align} {\label{log2}} I_2=\int_{B_{\frac{3 r}{2}}} |V| \frac{u^{p-1} \phi^p}{(u+t)^{p-1}} {\,\rm d}x& \leq \int_{B_{\frac{3 r}{2}}}|V| \phi^p {\,\rm d}x\leq\left\|V\right\|_{L^{q}(\Omega)} |B_{\frac{3r}{2}}|^{\frac{p}{d}-\frac{1}{q}} \left\| \phi \right\|_{L^{p^*}(B_{\frac{3r}{2}})}^p \nonumber\\ &\leq C(d,p) \|V\|_{L^{q}(\Omega)}\int_{B_{\frac{3 r}{2}}}|\nabla \phi|^p {\,\rm d}x\leq C \int_r^{\frac{3 r}{2}} \tau^{d-1-p} \mathrm{~d} \tau=C r^{d-p} . \end{align}\tag{25}\] Therefore, for \(\varepsilon=\frac{p-1}{2}\), using 24 , 25 we get \[\begin{align} {\label{log3}} I_1+I_2 \leq-C(p) \int_{B_{2 r}}|\nabla \log (u+t)|^p \phi^p {\,\rm d}x+C(d, p) r^{d-p}. \end{align}\tag{26}\] Further, following [8], for some \(C=C(d, p, s)\) we obtain \[\begin{align} {\label{log4}} I_s \leq-\frac{1}{C} \iint_{B_{2 r} \times B_{2 r}}\left|\log \left(\frac{u(x)+t}{u(y)+t}\right)\right|^p \phi(y)^p \mathrm{~d} \mu+C r^{d-p s}+C \frac{r^d}{t^{p-1} R^p} \operatorname{Tail}\left(u^{-} ; x_0, R\right)^{p-1}. \end{align}\tag{27}\] Therefore, the conclusion follows by combining 22 , 26 and 27 . ◻
In this section, we establish that the weak solution to 11 is locally bounded. We then prove the Tail estimate for the weak solution. The following lemma is required for local boundedness. For the proof, see [31].
Lemma 5 (Iteration Lemma). Let \((Y_j)_{j=0}^{\infty}\) be a sequence of positive real numbers such that \[\begin{align} Y_0\leq c_{0}^{-\frac{1}{\beta}}b^{-\frac{1}{\beta^2}} \text{ and } Y_{j+1}\leq c_0 b^{j} Y_j^{1+\beta}, \end{align}\] \(j=0,1,2,\dots\), for some constants \(c_0,b>1\) and \(\beta>0\). Then \(Y_j \rightarrow 0\) as \(j \rightarrow\infty\).
Proposition 5. Assume that \(u\) is a weak subsolution to 11 satisfying \(u \geq 0\) in \(B_R(x_0) \subset \Omega\). Let \(0<r\leq1\) be such that \(B_r(x_0) \subset B_R(x_0)\). Then there exists \(\zeta_2(d,p)>0\) such that for \(\|V\|_{L^q(\Omega)} \le \zeta_2\), \[\label{loc46bounded} \mathop{\mathrm{ess\,sup}}_{B_{\frac{r}{2}}(x_0)}u\leq \delta \theta^{\frac{1}{p-1}} \text{\rm Tail}(u^+;x_0,\frac{r}{2})+C(d,s,p)\delta^{\frac{(1-p)p^*}{p(p^*-p\sigma)}}\left(\fint_{B_r}u^{p\sigma}{\,\rm d}x\right)^{\frac{1}{p\sigma}},\qquad{(1)}\] where \(\delta\in(0,1], \sigma=\frac{(dp-d+p)q-d}{(dp-d+p)q-dp}\) and \(p\sigma<p^*\).
Proof. For \(r\in(0,1)\) and \(j=0,1,2, \cdots\), define \[r_j=\frac{r}{2}(1+2^{-j}),\,\overline{r}_j=\frac{r_j+r_{j+1}}{2},\,B_j=B_{r_j}(x_0),\, \text{ and } \,\overline{B}_j=B_{\overline{r}_j}(x_0).\] Let \(\{\phi_j\}_{j=1}^\infty\subset {\mathcal{C}}_c^{\infty}(\overline{B}_j)\) be a nonnegative sequence of cut-off functions such that \[0\leq \phi_j\leq 1\text{ in }\overline{B}_j,\,\phi_j\equiv1\text{ on }B_{j+1},\, |\nabla\phi_j|\leq \frac{2^{j+3}}{r}.\] For \(k,\overline{k}\geq0\), we denote \[k_j=k+(1-2^{-j})\overline{k},\,\overline{k}_j=\frac{k_j+k_{j+1}}{2},\, w_j=(u-k_j)_+,\, \text{ and } \,\overline{w}_j=(u-\overline{k}_j)_+.\] Using the energy estimate (Lemma 2), we obtain \[\label{caccioppoli-1} \begin{align} &\int_{B_j}\phi_j^p|\nabla \overline{w}_j|^p{\,\rm d}x+\theta\iint_{B_j^2}|\overline{w}_j(x)\phi_j(x)-\overline{w}_j(y)\phi_j(y)|^p\,{\rm d}\mu\\ &\leq C(p) \bigg(\int_{B_j}\overline{w}_j^p|\nabla\phi_j|^p{\,\rm d}x+\theta\iint_{B_j^2}\max\{\overline{w}_j(x),\overline{w}_j(y)\}^p \left| \phi_j(x)-\phi_j(y) \right|^p\,{\rm d}\mu\\ &\quad\qquad+\theta \,\underset{x\in\mathop{\mathrm{supp}}\phi}{\mathop{\mathrm{ess\,sup}}}\int_{\mathbb{R}^d\setminus B_j}\frac{\overline{w}_j(y)^{p-1}}{|x-y|^{d+sp}}{\,\rm d}y\cdot\int_{B_j}\overline{w}_j\phi_j^p{\,\rm d}x\bigg)+\int_{B_j}V(x)u^{p-1}\overline{w}_j\phi_j^p{\,\rm d}x\bigg). \end{align}\tag{28}\] Applying Hölder’s inequality, \[\begin{align} \label{lb-1} &\int_{B_j} V(x)u^{p-1}\overline{w}_j\phi_j^p{\,\rm d}x\le C(p) \left(\int_{B_j}\abs{V(x)}\overline{w}_j^p \phi_j^p {\,\rm d}x+\overline{k}_j^{p-1} \int_{B_j} |V(x)| \overline{w}_j\phi_j^p {\,\rm d}x\right) \nonumber\\ &\leq C(p) \left\|V\right\|_{L^{q}(\Omega)} |B_j|^{\frac{p}{d}-\frac{1}{q}} \left\| \overline{w}_j\phi_j \right\|_{L^{p^*}(B_j)}^p + \overline{k}_j^{p-1} \left\|V \phi_j^{p-1} \right\|_{L^{q}(B_j)} \left\|\overline{w}_j\phi_j\right\|_{L^{\frac{q}{q-1}}(B_j)} \nonumber\\ &\leq C(d,p) \|V\|_{L^{q}(\Omega)} \left(\left\|\overline{w}_j\phi_j\right\|_{L^{p^*}(B_j)}^p+\overline{k}_j^{p-1}\left\|\overline{w}_j\phi_j\right\|_{L^{\frac{q}{q-1}}(B_j)}\right). \end{align}\tag{29}\] Applying the generalized Hölder’s inequality \(\frac{1}{p_1} = \frac{\alpha}{p_2} + \frac{1-\alpha}{p_3}\) with the conjugate triplet \((p_1,p_2,p_3)\), where \[\begin{align} p_1=\frac{q}{q-1}, p_2= p^*, p_3=1, \text{ and } \alpha=\frac{p^*}{(p^*-1)q}, \end{align}\] and observing the fact that \[\begin{align} q>\frac{d}{p} \Longrightarrow \frac{q}{q-1} < \frac{d}{d-p} <p^* \Longrightarrow q > \frac{p^*}{p^*-1} \Longrightarrow \alpha< 1, \end{align}\] we now estimate \[\begin{align} \left\|\overline{w}_j\phi_j \right\|_{L^{\frac{q}{q-1}}(B_j)} \le \left\| \overline{w}_j \phi_j \right\|_{L^{p^*}(B_j)}^{\alpha} \left\|\overline{w}_j\phi_j\right\|_{L^1(B_j)}^{1-\alpha}. \end{align}\] Now, applying the Young’s inequality with coefficients \((\frac{p}{\alpha},\frac{p}{p-\alpha})\), we obtain \[\begin{align} \overline{k}_j^{p-1} \left\|\overline{w}_j\phi_j\right\|_{L^{p^*}(B_j)}^\alpha\,\left\|\overline{w}_j\phi_j\right\|_{L^{1}(B_j)}^{1-\alpha} \leq \|\overline{w}_j\phi_j\|_{L^{p^*}(B_j)}^p+\overline{k}_j^{\frac{p(p-1)}{p-\alpha}}\|\overline{w}_j\phi_j\|_{L^{1}(B_j)}^{\frac{p(1-\alpha)}{p-\alpha}}. \end{align}\] Using the Sobolev embedding \(W^{1,p}_0(B_j) \hookrightarrow L^{p^*}(B_j)\) and the fact that \(\overline{w}_j\phi_j \in W^{1,p}_0(B_j)\), we get \[\begin{align} \|\overline{w}_j\phi_j\|_{L^{p^*}(B_j)}^p \le C(d,p) \@ifstar{\norm}{\norm*}{\nabla(\overline{w}_j\phi_j)}_{L^{p}(B_j)}^p \le C(d,p) \left( \int_{B_j}\phi_j^p|\nabla \overline{w}_j|^p {\,\rm d}x+ \int_{B_j}\overline{w}_j^p|\nabla\phi_j|^p{\,\rm d}x\right). \end{align}\] Therefore, noting 29 and choosing \(\zeta_2 = \zeta_2(d,p) >0\) small enough, we get from 28 , \[\label{caccioppoli-2} \begin{align} &\int_{B_j}\phi_j^p|\nabla \overline{w}_j|^p {\,\rm d}x+\theta\iint_{B_j^2}|\overline{w}_j(x)\phi_j(x)-\overline{w}_j(y)\phi_j(y)|^p\,{\rm d}\mu\\ &\leq C\bigg(\int_{B_j}\overline{w}_j^p|\nabla\phi_j|^p{\,\rm d}x+\theta\iint_{B_j^2}\max\{\overline{w}_j(x),\overline{w}_j(y)\}^p|\phi_j(x)-\phi_j(y)|^p\,{\rm d}\mu\\ &\quad\qquad+ \theta \, \underset{x\in\mathop{\mathrm{supp}}\phi}{\text{ess sup}}\int_{\mathbb{R}^d\setminus B_j}\frac{\overline{w}_j(y)^{p-1}}{|x-y|^{d+sp}}{\,\rm d}y\cdot\int_{B_j}\overline{w}_j\phi_j^p{\,\rm d}x+\overline{k}_j^{\frac{p(p-1)}{p-\alpha}}\|\overline{w}_j\phi_j\|_{L^{1}(B_j)}^{\frac{p(1-\alpha)}{p-\alpha}}\bigg). \end{align}\tag{30}\] Again, using the Sobolev inequality, 30 yields \[\begin{align} \label{caccioppoli-3} &\left(\fint_{B_j}|\overline{w}_j \phi_j|^{p^*}{\,\rm d}x\right)^{\frac{p}{p^*}}\leq Cr^{p-d}\|\nabla(\overline{w}_j\phi_j)\|_{L^p(B_j)}^p\leq Cr^{p-d}\left(\int_{B_j}\phi_j^p|\nabla \overline{w}_j|^p{\,\rm d}x+\int_{B_j}\overline{w}_j^p|\nabla\phi_j|^p{\,\rm d}x\right) \nonumber\\ &\leq Cr^{p-d}\bigg(\int_{B_j}\overline{w}_j^p|\nabla\phi_j|^p{\,\rm d}x+\theta\iint_{B_j^2}\max\{\overline{w}_j(x),\overline{w}_j(y)\}^p|\phi_j(x)-\phi_j(y)|^p\,{\rm d}\mu\nonumber\\ &+\theta \,\underset{x\in\mathop{\mathrm{supp}}\phi}{\text{ess sup}}\int_{\mathbb{R}^d\setminus B_j}\frac{\overline{w}_j(y)^{p-1}}{|x-y|^{d+sp}}{\,\rm d}y\cdot\int_{B_j}\overline{w}_j\phi_j^p{\,\rm d}x+\overline{k}_j^{\frac{p(p-1)}{p-\alpha}}\|\overline{w}_j\phi_j\|_{L^1(B_j)}^{\frac{p(1-\alpha)}{p-\alpha}}\bigg), \end{align}\tag{31}\] where \(C=C(d,p,s)\). Using \(|\nabla\phi_j|\leq \frac{2^{j+3}}{r}\), observe that \[\begin{align} r^{p-d} \int_{B_j}\overline{w}_j^p|\nabla\phi_j|^p{\,\rm d}x\le C(d,p) 2^{jp} \fint_{B_j} {w}_j^p {\,\rm d}x. \end{align}\] Further, we estimate \[\begin{align} &\theta r^p \fint_{B_j} \int_{B_j}\max\{\overline{w}_j(x),\overline{w}_j(y)\}^p|\phi_j(x)-\phi_j(y)|^p\,{\rm d}\mu\le C\theta 2^{jp} \fint_{B_j} w_j^p(y) \left( \int_{B_j} \frac{{\,\rm d}x}{\abs{x-y}^{d+ps-p}} \right) {\,\rm d}y\\ & \le \frac{C\theta 2^{jp}}{p(1-s)} r^{p-ps}\fint_{B_j} w_j^p(x) {\,\rm d}x\le C2^{jp}\fint_{B_j} w_j^p(x) {\,\rm d}x. \end{align}\] Proceeding similarly as in the proof of [30], we get \[\begin{align} & \theta r^{p-d} \, \underset{x\in\mathop{\mathrm{supp}}\phi}{\mathop{\mathrm{ess\,sup}}}\int_{\mathbb{R}^d\setminus B_j}\frac{\overline{w}_j(y)^{p-1}}{|x-y|^{d+sp}}{\,\rm d}y\cdot\int_{B_j}\overline{w}_j\phi_j^p{\,\rm d}x\\ &\le C(d,s,p) 2^{j(d+sp+p-1)} \theta \left( \frac{\text{Tail}(u^+;x_0,\frac{r}{2})}{\overline{k}} \right)^{p-1} \fint_{B_j} w_j^p {\,\rm d}x. \end{align}\] Also, observe that \[\fint_{B_j}w_j^p{\,\rm d}x\leq \left(\fint_{B_j}w_j^{p\sigma}\right)^{\frac{1}{\sigma}}.\] Next, we estimate the last integral of 31 . Define \(\sigma:=\frac{p-\alpha}{p(1-\alpha)}>1\). Then \(p\sigma\in(p,p^*)\). Using the relations \[\begin{align} \frac{p(p-1)}{p- \alpha} = p - \frac{1}{\sigma}, \overline{w}_j\leq(\overline{k}_j-k_j)^{1-p\sigma}w_j^{p\sigma}, \text{ and } \overline{k}_j\leq k+\overline{k}, \end{align}\] we get \[\begin{align} \overline{k}_j^{\frac{p(p-1)}{p-\alpha}}\|\overline{w}_j\phi_j\|_{L^1(B_j)}^{\frac{1}{\sigma}}\leq\left(\frac{k+\overline{k}}{\overline{k}_j-k_j}\right)^{p-\frac{1}{\sigma}}\left(\int_{B_j}w_j^{p\sigma}\right)^{\frac{1}{\sigma}} \le \left(2^{j+2} \frac{k+\overline{k}}{\overline{k}}\right)^{p-\frac{1}{\sigma}} r^{\frac{d}{\sigma}}\left(\fint_{B_j}w_j^{p\sigma}\right)^{\frac{1}{\sigma}}. \end{align}\] Therefore, from 31 , there exists \(C=C(d,p,s)\) such that for every large \(j\), \[\begin{align} \left(\fint_{\overline{B}_j}|\overline{w}_j\phi_j|^{p^*}\right)^{\frac{p}{p^*}}&\leq C \left[2^{jp}+2^{j(d+sp+p-1)} \theta \left( \frac{\text{Tail}(u^+;x_0,\frac{r}{2})}{\overline{k}} \right)^{p-1} +r^{p-d+\frac{d}{\sigma}}\left(2^{j+2}\frac{k+\overline{k}}{\overline{k}}\right)^{p-\frac{1}{\sigma}}\right] \\ &\quad \left(\fint_{B_j}w_j^{p\sigma}\right)^{\frac{1}{\sigma}} \\ & \le C 2^{j(d+sp+p-1)} \left[\theta \left( \frac{\text{Tail}(u^+;x_0,\frac{r}{2})}{\overline{k}} \right)^{p-1} +r^{p-d+\frac{d}{\sigma}}\left(\frac{k}{\overline{k}}\right)^{p-\frac{1}{\sigma}}+1\right] \left(\fint_{B_j}w_j^{p\sigma}\right)^{\frac{1}{\sigma}}, \end{align}\] where using the definition of \(\sigma\), we observe that \(p-d+\frac{d}{\sigma}\geq 0\). For \(\delta\in(0,1]\), we let \(\overline{k}\) such that \[\begin{align} \theta \left( \frac{\text{Tail}(u^+;x_0,\frac{r}{2})}{\overline{k}} \right)^{p-1} \le \delta^{1-p} \Longleftrightarrow \overline{k} \ge \delta \theta^{\frac{1}{p-1}}\text{Tail}(u^+;x_0,\frac{r}{2}), \end{align}\] and \[\begin{align} r^{p-d+\frac{d}{\sigma}}\left(\frac{k}{\overline{k}}\right)^{p-\frac{1}{\sigma}} \le \delta^{1-p} \Longleftrightarrow \overline{k} \ge kr^{\left(p-d+\frac{d}{\sigma} \right)\left( \frac{p-\alpha}{p(p-1)}\right)} \delta^{\frac{p-\alpha}{p}}. \end{align}\] In view of the above inequalities, we take \[\overline{k}\geq \delta \theta^{\frac{1}{p-1}}\text{Tail}(u^+;x_0,\frac{r}{2})+kr^{\left(p-d+\frac{d}{\sigma} \right)\left( \frac{p-\alpha}{p(p-1)}\right)} \delta^{\frac{p-\alpha}{p}},\] so that \[\begin{align} \label{lb-3} \left(\fint_{\overline{B}_j}|\overline{w}_j\phi_j|^{p^*}\right)^{\frac{p}{p^*}}&\leq C 2^{j(d+sp+p-1)}\delta^{1-p}\left(\fint_{B_j}w_j^{p\sigma}\right)^{\frac{1}{\sigma}}. \end{align}\tag{32}\] We also have \[\begin{align} \label{lb-4} \left(\fint_{\overline{B}_j}|\overline{w}_j\phi_j|^{p^*}\right)^{\frac{p}{p^*}}&\geq (k_{j+1}-\overline{k}_j)^{\frac{p(p^*-p\sigma)}{p^*}}\left(\fint_{B_{j+1}}w_{j+1}^{p\sigma}\right)^{\frac{p}{p^*}} \nonumber\\ &= \left(\frac{\overline{k}}{2^{j+2}}\right)^{\frac{p(p^*-p\sigma)}{p^*}}\left(\fint_{B_{j+1}}w_{j+1}^{p\sigma}\right)^{\frac{p}{p^*}}. \end{align}\tag{33}\] Denote \(Y_j:=\displaystyle \left(\fint_{B_j}w_j^{p\sigma}\right)^{\frac{1}{p\sigma}}.\) Then using 32 and 33 , there exists \(C=C(d,p,s)>1\) such that \[\begin{align} \left(\frac{\overline{k}}{2^{j}}\right)^{\frac{p(p^*-p\sigma)}{p^*}}Y_{j+1}^{\frac{p^2\sigma}{p^*}}\leq C 2^{j(d+sp+p-1)} \delta^{1-p} Y_j^p, \end{align}\] which implies \[\begin{align} \frac{Y_{j+1}}{\overline{k}}\leq C \delta^{\frac{p^*(1-p)}{p^2 \sigma}} \tilde{C}^j\left(\frac{Y_j}{\overline{k}}\right)^{1+\beta}, \end{align}\] where using the fact \(p\sigma<p^*\), we see that \(\beta:=\frac{p^*}{p\sigma}-1>0\) and \(\tilde{C}:=2^{\left(\frac{d+sp+p-1}{p}+\frac{p^*-p\sigma}{p^*}\right)\frac{p^*}{p\sigma}}>1.\) Finally, we choose \[\overline{k}:=\delta \theta^{\frac{1}{p-1}}\text{Tail}(u^+;x_0,\frac{r}{2})+kr^{\left(p-d+\frac{d}{\sigma} \right)\left( \frac{p-\alpha}{p(p-1)}\right)} \delta^{\frac{p-\alpha}{p}}+\delta^{\frac{(1-p)p^*}{p(p^*-p\sigma)}}C^{\frac{1}{\beta}}\tilde{C}^{\frac{1}{\beta^2}}Y_0,\] so that \[\frac{Y_0}{\overline{k}}\leq \delta^{\frac{p^*(p-1)}{p^2\sigma\beta}}C^{-\frac{1}{\beta}} \tilde{C}^{-\frac{1}{\beta^2}} = \left( C \delta^{\frac{p^*(1-p)}{p^2\sigma}} \right)^{-\frac{1}{\beta}} \tilde{C}^{-\frac{1}{\beta^2}}.\] Now we take \(c_0 = C\delta^{\frac{p^*(1-p)}{p^2\sigma}}\) and \(b=\tilde{C}\) in Lemma 5. Therefore, applying the iteration lemma (Lemma 5), we get \(Y_j\to0\text{ as }j\to\infty.\) Hence \[\begin{align} \sup_{B_{\frac{r}{2}}(x_0)}(u-k)^+\leq \overline{k} & \leq\delta \theta^{\frac{1}{p-1}}\text{Tail}(u^+;x_0,\frac{r}{2})+kr^{\left(p-d+\frac{d}{\sigma} \right)\left( \frac{p-\alpha}{p(p-1)}\right)} \delta^{\frac{p-\alpha}{p}} \\ & +\delta^{\frac{(1-p)p^*}{p(p^*-p\sigma)}}C^{\frac{1}{\beta}}\tilde{C}^{\frac{1}{\beta^2}}\left(\fint_{B_r}u^{p\sigma}\right)^{\frac{1}{p\sigma}}. \end{align}\] Observe that \(C^{\frac{1}{\beta}}\tilde{C}^{\frac{1}{\beta^2}}=C(d,s,p)\). Now, choosing \(k=0\) yields ?? . ◻
In the following lemma, we state the Tail estimate for weak solution to 11 .
Lemma 6. Assume that \(u\) is a weak supersolution to 11 satisfying \(u \geq 0\) in \(B_R(x_0) \subset \Omega\). Let \(0<r\leq1\) be such that \(B_r(x_0) \subset B_R(x_0)\). Then \[\label{tailest} \text{\rm Tail}(u^{+};x_0,r)\leq C(d,p,s)\left( \theta^{\frac{1}{1-p}} \mathop{\mathrm{ess\,sup}}_{B_r(x_0)}\,u+\Big(\frac{r}{R}\Big)^\frac{p}{p-1}\text{\rm Tail}(u^{-};x_0,R)\right).\tag{34}\]
Proof. Let \(M=\underset{B_r}{\mathop{\mathrm{ess\,sup}}}\,u\) and \(\phi\in {\mathcal{C}}_c^{\infty}(B_r)\) be a cut-off function such that \(0\leq\phi\leq 1\) in \(B_r\), \(\phi=1\) in \(B_{\frac{r}{2}}(x_0)\) and \(|\nabla\phi|\leq\frac{8}{r}\) in \(B_r\). By letting \(w=u-2M\) and choosing \(\phi=w\phi^p\) as a test function we obtain \[\begin{align} \label{tailtest} \int_{B_r} V(x)u^{p-1}w\phi^p{\,\rm d}x&=\int_{B_r}|\nabla u|^{p-2}\nabla u\cdot\nabla(w\phi^p){\,\rm d}x\nonumber\\ &\qquad+\theta\int_{B_r}\int_{B_r}\mathcal{A}(u(x,y))(w(x)\phi(x)^p-w(y)\phi(y)^p)\,{\rm d}\mu\nonumber\\ &\qquad+2\theta\int_{B_r}\int_{\mathbb{R}^d\setminus B_r}\mathcal{A}(u(x,y))w(x)\phi(x)^p\,{\rm d}\mu\nonumber\\ &=I_1+I_2+I_3. \end{align}\tag{35}\] By Young’s inequality, \[\begin{align} |\nabla w|^{p-2}\nabla w\cdot\nabla(w\phi^p) &=|\nabla w|^p \phi^p+p\phi^{p-1}w|\nabla w|^{p-2}\nabla w\cdot\nabla\phi\\ &\geq\frac{1}{2}|\nabla w|^p\phi^p-C(p)|u|^p|\nabla\phi|^p -C(p)M^p|\nabla\phi|^p, \end{align}\] holds in \(B_r\). By the properties of \(\phi\), we have \[\label{tailestI1} I_1=\int_{B_r}|\nabla u|^{p-2}\nabla u\cdot\nabla(w\phi^p) {\,\rm d}x\geq -C(p)M^p r^{-p}|B_r|.\tag{36}\] In view of \((4.11)\) and \((4.9)\) in [8] and using the fact that \(r\in(0,1)\), there exists \(C=C(d,p,s)>0\) such that \[\begin{align} \label{tailestI2} I_2&=\theta\int_{B_r}\int_{B_r}\mathcal{A}(u(x,y))\left(w(x)\phi(x)^p-w(y)\phi(y)^p\right) \,{\rm d}\mu\geq -C \theta M^p r^{-p}|B_r|, \end{align}\tag{37}\] and \[\begin{align} \label{tailestI3} I_3&=2\theta\int_{B_r}\int_{\mathbb{R}^d\setminus B_r}\mathcal{A}(u(x,y))w(x)\phi(x)^p \,{\rm d}\mu \geq C \theta M r^{-p}\text{\rm Tail}(u^{+};x_0,r)^{p-1}|B_r|\nonumber\\ &\qquad-C \theta MR^{-p}\text{\rm Tail}(u^{-};x_0,R)^{p-1}|B_r|- C \theta M^p r^{-p}|B_r|. \end{align}\tag{38}\] Further, \[\begin{align} \label{tailestI4} \int_{B_r} |V(x)|u^{p-1}|u-2M|\phi^p{\,\rm d}x\le M^p \int_{B_r} |V(x)| {\,\rm d}x\le M^p \@ifstar{\norm}{\norm*}{V}_{L^q(B_r)} |B_r|^{\frac{q-1}{q}}, \end{align}\tag{39}\] Combining 36 39 , we get from 35 , \[\begin{align} \text{\rm Tail}(u^{+};x_0,r)\leq C(d,p,s)\left(M\theta^{\frac{1}{1-p}} + \Big(\frac{r}{R}\Big)^\frac{p}{p-1}\text{\rm Tail}(u^{-};x_0,R) + M\theta^{\frac{1}{1-p}} \left[ \@ifstar{\norm}{\norm*}{V}_{L^q} r^p |B_r|^{-\frac{1}{q}} \right]^{\frac{1}{p-1}} \right), \end{align}\] where \(r^p|B_r(x_0)|^{\frac{-1}{q}}=(\omega_d)^{-\frac{1}{q}} r^{p-\frac{d}{q}}\leq C(d)\) using the fact that \(r \le 1\) and \(q>\frac{d}{p}\). Thus, 34 holds. ◻
This section contains the proof of the Harnack inequalities. We start with the following lemma that proves the expansion of positivity for a weak supersolution to 11 .
Lemma 7. Assume that \(u\) is a weak supersolution to 11 satisfying \(u \geq 0\) in \(B_R(x_0) \subset \Omega\). Let \(0<r\leq1\) be such that \(B_r(x_0) \subset B_{\frac{R}{16}}(x_0)\). Assume \(k \geq 0\) and there exists \(\tau \in(0,1]\) such that \[\begin{align} {\label{expan1}} \left|B_r \cap\{u \geq k\}\right| \geq \tau\left|B_r(x_0\right)|. \end{align}\tag{40}\] Then there exists \(\zeta_3(d,p)>0\) such that for \(\|V\|_{L^q(\Omega)} \le \zeta_3\), \[\begin{align} {\label{expan1461}} \underset{B_{4 r}\left(x_0\right)}{\operatorname{\mathop{\mathrm{ess\,inf}}}} u \geq \delta k-\theta^{\frac{1}{p-1}}\left(\frac{r}{R}\right)^{\frac{p}{p-1}} \operatorname{Tail}\left(u^{-} ; x_0, R\right), \end{align}\tag{41}\] where \(\delta=\delta(d, p, s, \tau) \in \left(0, \frac{1}{4}\right)\).
Proof. Our proof includes two steps.
Step 1. Let \(\varepsilon>0\). Under the assumption in 40 , we claim that there exists \(C_1=C_1(d, p, s)\) such that
\[\begin{align}
{\label{expan2}}
\left|B_{6 r}(x_0) \cap\left\{u \leq 2 \delta k-\frac{1}{2}\theta^{\frac{1}{p-1}}\left(\frac{r}{R}\right)^{\frac{p}{p-1}} \operatorname{Tail}\left(u^{-} ; x_0, R\right)-\varepsilon\right\}\right| \leq \frac{C_1}{\tau \log \frac{1}{2 \delta}}\left|B_{6
r}(x_0)\right|,
\end{align}\tag{42}\] for every \(\delta \in\left(0, \frac{1}{4}\right)\). Consider a cut-off function \(\phi \in {\mathcal{C}}_c^{\infty}(B_{7 r}(x_0))\) such that \(0 \leq \phi \leq 1\) in \(B_{7 r}(x_0), \phi=1\) in \(B_{6 r}(x_0)\) and \(|\nabla \phi| \leq \frac{8}{r}\) in \(B_{7 r}(x_0)\). Define \(v:=u+t_\varepsilon\), where \[t_\varepsilon=\frac{1}{2}\theta^{\frac{1}{p-1}}\left(\frac{r}{R}\right)^{\frac{p}{p-1}}
\operatorname{Tail}\left(u^{-} ; x_0, R\right)+\varepsilon.\] As \(u\) is a weak supersolution, we choose \(\phi=v^{1-p} \phi^p\) as a test function and estimate like Lemma 4 to obtain \[\begin{align}
{\label{expan3}}
&\int_{B_{6r(x_0)}}|\nabla \log (u+t_{\epsilon})|^p {\,\rm d}x+\theta\iint_{B_{6r(x_0)} \times B_{6r(x_0)}}\left|\log \left(\frac{u(x)+t_{\epsilon}}{u(y)+t_{\epsilon}}\right)\right|^p \,{\rm d}\mu\nonumber\\&\leq C r^d\left(r^{-p}+\theta r^{-p
s}+\theta t_{\epsilon}^{1-p} R^{-p} \operatorname{Tail}\left(u^{-} ; x_0, R\right)^{p-1}\right)\leq Cr^{d-p},
\end{align}\tag{43}\] where \(C=C(d, p, s)\). For \(\delta \in\left(0, \frac{1}{4}\right)\), we denote \[w=\left(\min \left\{\log \frac{1}{2 \delta},
\log \frac{k+t_\varepsilon}{v}\right\}\right)^{+} .\] By 43 , we have \[\begin{align}
{\label{expan4}}
\int_{B_{6 r}(x_0)}|\nabla w|^p {\,\rm d}x\leq \int_{B_{6 r}(x_0)}|\nabla \log v|^p {\,\rm d}x\leq C r^{d-p}.
\end{align}\tag{44}\] From 44 , by Hölder’s inequality and Poincaré inequality, we obtain \[\begin{align}
{\label{expan5}}
\int_{B_{6 r}(x_0)}\left|w-(w)_{B_{6 r}(x_0)}\right| {\,\rm d}x& \leq Cr^{\frac{d}{p^\prime}}\left(\int_{B_{6 r}(x_0)}\left|w-(w)_{B_{6 r}(x_0)}\right|^p {\,\rm d}x\right)^{\frac{1}{p}} \nonumber\\&\leq Cr^{1+\frac{d}{p^{\prime}}}\left(\int_{B_{6
r}\left(x_0\right)}|\nabla w|^p {\,\rm d}x\right)^{\frac{1}{p}} \leq C\left|B_{6 r}(x_0)\right|,
\end{align}\tag{45}\] where \(p^{\prime}=\frac{p}{p-1}\) and \((w)_{B_{6 r}(x_0)}=\fint_{B_{6 r}\left(x_0\right)} v {\,\rm d}x\). We observe that \(\{w=0\}=\left\{v \geq k+t_\varepsilon\right\}=\{u \geq k\}\). By the assumption 40 , it follows that \[\begin{align}
{\label{expan6}} \left|B_{6 r}(x_0) \cap\{w=0\}\right| \geq \frac{\tau}{6^d}\left|B_{6 r}\left(x_0\right)\right|.
\end{align}\tag{46}\] Following the proof of [8] and using 46 , we obtain
\[\begin{align}
{\label{expan7}}
\log \frac{1}{2 \delta} & =\frac{1}{\left|B_{6 r}(x_0) \cap\{w=0\}\right|} \int_{B_{6 r}(x_0) \cap\{w=0\}}\left(\log \frac{1}{2 \delta}-w(x)\right) {\,\rm d}x\leq \frac{6^d}{\tau}\left(\log \frac{1}{2 \delta}-(w)_{B_{6 r}}\right).
\end{align}\tag{47}\] Now integrating 47 over the set \(B_{6 r}(x_0) \cap\left\{w=\log \frac{1}{2 \delta}\right\}\) and using 45 , there exists \(C_1=C_1(d, p, s)\) such that \[\left|\left\{w=\log \frac{1}{2 \delta}\right\} \cap B_{6 r}(x_0)\right| \log \frac{1}{2 \delta} \leq \frac{6^d}{\tau} \int_{B_{6 r}(x_0)}\left|w-(w)_{B_{6
r}(x_0)}\right| {\,\rm d}x\leq \frac{C_1}{\tau}\left|B_{6 r}(x_0)\right| .\] Hence, for any \(\delta \in\left(0, \frac{1}{4}\right)\), we have \[\left|B_{6 r}(x_0) \cap\left\{v \leq 2
\delta\left(k+t_\varepsilon\right)\right\}\right| \leq \frac{C_1}{\tau} \frac{1}{\log \frac{1}{2 \delta}}\left|B_{6 r}\left(x_0\right)\right| .\] This implies 42 .
Step 2. We claim that, for every \(\varepsilon>0\), there exists a constant \(\delta=\delta(d, p, s, \tau) \in\left(0, \frac{1}{4}\right)\) such that
\[\begin{align}
{\label{expan8}}
\mathop{\mathrm{ess\,inf}}_{B_{4 r}(x_0)} u \geq \delta k-\theta^{\frac{1}{p-1}}\left(\frac{r}{R}\right)^{\frac{p}{p-1}} \operatorname{Tail}\left(u^{-} ; x_0, R\right)-2 \varepsilon.
\end{align}\tag{48}\] As a consequence of 48 , the estimate 41 follows. To prove 48 , without loss of generality, we may assume that
\[\begin{align}
{\label{expan9}} \delta k \geq\theta^{\frac{1}{p-1}}\left(\frac{r}{R}\right)^{\frac{p}{p-1}} \operatorname{Tail}\left(u^{-} ; x_0, R\right)+2 \varepsilon.
\end{align}\tag{49}\] Otherwise 48 holds true, since \(u \geq 0\) in \(B_r\). Let \(\rho \in[r, 6 r]\) and \(\phi \in {\mathcal{C}}_c^{\infty}(B_\rho(x_0))\) be a cut-off function such that \(0 \leq \phi \leq 1\) in \(B_\rho(x_0)\). For any \(l
\in(\delta k, 2 \delta k)\), we denote \(w=(l-u)^{+}\). From Lemma 2, for \(C=C(d, p, s)\),
we obtain \[\begin{align}
{\label{expan10}} \fint_{B_{\rho}} \phi^{p} |\nabla w|^{p} {\,\rm d}x &\le C \Bigg( \fint_{B_{\rho}} w^{p} |\nabla \phi|^{p} {\,\rm d}x+ \fint_{B_{\rho}}| V(x)| u^{p-1} w \phi^p {\,\rm d}x\nonumber \\ &+ \theta\fint_{B_\rho}\int_{B_\rho}
\max\{w(x), w(y)\}^{p} |\phi(x)-\phi(y)|^{p} \,{\rm d}\mu\nonumber \\ &+ \theta\mathop{\mathrm{ess\,sup}}_{x \in \text{supp} (\phi)} \int_{\mathbb{R}^d\setminus B_{\rho}} \frac{w(y)^{p-1}}{|x-y|^{d+ps}} {\,\rm d}y\cdot \fint_{B_{\rho}} w\phi^{p} {\,\rm
d}x\Bigg),
\end{align}\tag{50}\] We will apply Lemma 5 to conclude the proof. For \(j=0,1,2, \ldots\), we denote \[\begin{align}
l=k_j=\delta k+2^{-j-1} \delta k, \quad \rho=\rho_j=4 r+2^{1-j} r, \quad \hat{\rho}_j=\frac{\rho_j+\rho_{j+1}}{2}.
\end{align}\] Then \(l \in(\delta k, 2 \delta k), \rho_j, \hat{\rho_j} \in(4 r, 6 r)\) and \[\begin{align}
{\label{obser}}
k_j-k_{j+1}=2^{-j-2} \delta k \geq 2^{-j-3} k_j,
\end{align}\tag{51}\] for every \(j\). Set \(B_j=B_{\rho_j}\left(x_0\right), \hat{B}_j=B_{\hat{\rho}_j}\left(x_0\right)\) and we observe that \[\begin{align}
w_j=\left(k_j-u\right)^{+} \geq 2^{-j-3} k_j \chi_{\left\{u<k_{j+1}\right\}}.
\end{align}\] We now consider a sequence of cut-off functions \(\left(\phi_j\right)_{j=0}^{\infty} \subset {\mathcal{C}}_c^{\infty}(\hat{B}_j)\) such that \(0 \leq \phi_j \leq 1\) in
\(\hat{B}_j, \phi_j=1\) in \(B_{j+1}\) and \(\left|\nabla \phi_j\right| \leq \frac{2^{j+3}}{r}\). We choose \(\phi=\phi_j,
w=w_j\) in 50 to have \[\begin{align}
{\label{expan101}} \fint_{B_{j}} \phi_j^{p} |\nabla w_j|^{p} {\,\rm d}x &\le C \Bigg( \fint_{B_{j}} w_j^{p} |\nabla \phi_j|^{p} {\,\rm d}x+ \fint_{B_{j}}| V(x)| u^{p-1} w _j\phi_j^p {\,\rm d}x\nonumber \\ &+ \theta\fint_{B_j}\int_{B_j}
\max\{w_j(x), w_j(y)\}^{p} |\phi_j(x)-\phi_j(y)|^{p} \,{\rm d}\mu\nonumber \\ &+ \theta\mathop{\mathrm{ess\,sup}}_{x \in \text{supp} (\phi_j)} \int_{\mathbb{R}^d\setminus B_{j}} \frac{w_j(y)^{p-1}}{|x-y|^{d+ps}} {\,\rm d}y\cdot \fint_{B_{j}}
w_j\phi_j^{p} {\,\rm d}x\Bigg)=:J_1+J_2+J_3+J_4.
\end{align}\tag{52}\] We estimate each term present in the right-hand side of 52 . Using the properties of \(\phi_j\), we have \[\begin{align}
{\label{J951ex}}
J_1=\fint_{B_{j}} w_j^{p} |\nabla \phi_j|^{p} {\,\rm d}x\leq C 2^{jp}r^{-p}\fint_{B_{j}} w_j^{p} {\,\rm d}x.
\end{align}\tag{53}\] For the term \(J_2\), recall that \(V\in L^q(\Omega)\), where \(q\) is described in 2 , and
estimate as \[\begin{align}
{\label{expan13}}
\int_{B_{j}}| V(x)| u^{p-1} w_j \phi_j^p {\,\rm d}x=\int_{B_{j}}| V(x)| u^{p-1} (k_j-u)^+ \phi_j^p {\,\rm d}x&\leq k_j^{p-1}\int_{B_{j}}| V(x)| (k_j-u)^+ \phi_j^p {\,\rm d}x\nonumber\\&\leq {k}_j^{p-1} \left\|V \phi_j^{p-1} \right\|_{L^{q}(B_j)}
\left\|{w}_j\phi_j\right\|_{L^{\frac{q}{q-1}}(B_j)} \nonumber\\&\leq {k}_j^{p-1} \left\|V \right\|_{L^{q}(B_j)} \left\|{w}_j\phi_j\right\|_{L^{\frac{q}{q-1}}(B_j)}.
\end{align}\tag{54}\] Now, proceeding as in Proposition 5 we obtain \[\begin{align}
{k}_j^{p-1} \left\|{w}_j\phi_j\right\|_{L^{\frac{q}{q-1}}(B_j)} \leq \|{w}_j\phi_j\|_{L^{p^*}(B_j)}^p+{k}_j^{\frac{p(p-1)}{p-\alpha}}\|{w}_j\phi_j\|_{L^{1}(B_j)}^{\frac{p(1-\alpha)}{p-\alpha}},
\end{align}\] where \(\alpha< 1\). By \(W^{1,p}_0(B_j) \hookrightarrow L^{p^*}(B_j)\) and the fact that \({w}_j\phi_j \in W^{1,p}_0(B_j)\), we get
\[\begin{align} \|{w}_j\phi_j\|_{L^{p^*}(B_j)}^p \le C(d,p) \left( \int_{B_j}\phi_j^p|\nabla {w}_j|^p {\,\rm d}x+ \int_{B_j}{w}_j^p|\nabla\phi_j|^p{\,\rm d}x\right).
\end{align}\] Observe that \(w_j\) is supported in \(\{ u<k_j\}\) and \(w_j \le k_j\) a.e. in \(B_j \cap \{
u<k_j\}\). Therefore, choosing \(\zeta_3 = \zeta_3(d,p) >0\) small enough so that \(\|V\|_{L^q(\Omega)}C(d,p)\leq \frac{1}{2}\), and noting that \(\rho_j\in(4r,6r)\), we use the properties of \(\phi_j\) and 54 , to get \[\begin{align}
{\label{J952ex}}
J_2&=\fint_{B_{j}}| V(x)| u^{p-1} w_j \phi_j^p {\,\rm d}x\nonumber\\&\leq \frac{1}{2}\fint_{B_j}\phi_j^p|\nabla {w}_j|^p {\,\rm d}x+\fint_{B_{j}} w_j^{p} |\nabla \phi_j|^{p} {\,\rm d}x+C r^{-d}
\|V\|_{L^q(\Omega)}{k}_j^{\frac{p(p-1)}{p-\alpha}}\|{w}_j\phi_j\|_{L^{1}(B_j)}^{\frac{p(1-\alpha)}{p-\alpha}}\nonumber\\&\leq \frac{1}{2}\fint_{B_j}\phi_j^p|\nabla {w}_j|^p {\,\rm d}x+\fint_{B_{j}} w_j^{p} |\nabla \phi_j|^{p} {\,\rm d}x+C
r^{-d}\|V\|_{L^q(\Omega)}{k}_j^{\frac{p(p-1)}{p-\alpha}+\frac{p(1-\alpha)}{p-\alpha}}\|\phi_j\|_{L^{1}(B_j \cap\left\{u<k_j\right\})}^{\frac{p(1-\alpha)}{p-\alpha}}\nonumber\\&\leq \frac{1}{2}\fint_{B_j}\phi_j^p|\nabla {w}_j|^p {\,\rm
d}x+\fint_{B_{j}} w_j^{p} |\nabla \phi_j|^{p} {\,\rm d}x+C r^{-d}\|V\|_{L^q(\Omega)}k_j^p\left|B_j \cap\left\{u<k_j\right\}\right|^{\frac{1}{\sigma}}\nonumber\\&\leq \frac{1}{2}\fint_{B_j}\phi_j^p|\nabla {w}_j|^p {\,\rm d}x+C
2^{jp}r^{-p}\fint_{B_{j}} w_j^{p}+C r^{\frac{d}{\sigma}-d}\|V\|_{L^q(\Omega)}k_j^p\left(\frac{\left|B_j \cap\left\{u<k_j\right\}\right|}{|B_j|}\right)^{\frac{1}{\sigma}},
\end{align}\tag{55}\] where \(\sigma=\frac{(dp-d+p)q-d}{(dp-d+p)q-dp}>1\) and \(C=C(d,p,s)\). In the second last line of 55 we have used the fact
\(\frac{p(p-1)}{p- \alpha} = p - \frac{1}{\sigma}\). Further, we estimate \[\begin{align}
{\label{J953ex}} J_3=\theta\fint_{B_j}\int_{B_j} \max\{w_j(x), w_j(y)\}^{p} |\phi_j(x)-\phi_j(y)|^{p} \,{\rm d}\mu& \le C 2^{jp} r^{-p} \fint_{B_j} w_j^p(x) \left( \int_{B_j} \frac{{\,\rm d}y}{\abs{x-y}^{d+ps-p}} \right) {\,\rm d}x\nonumber\\ & \le
\frac{C2^{jp}r^{-p}r^{p-ps}}{p(1-s)} \fint_{B_j} w_j^p {\,\rm d}x\nonumber\\& \le C2^{jp}r^{-p}\fint_{B_j} w_j^p {\,\rm d}x,
\end{align}\tag{56}\] where \(C=C(p,s)\). To estimate \(J_4\), we observe that, for any \(x \in \operatorname{supp} \phi_j \subset \hat{B}_j\)
and \(y \in \mathbb{R}^d \backslash B_j\), we have \[\frac{\left|y-x_0\right|}{|y-x|}=\frac{\left|y-x+x-x_0\right|}{|y-x|} \leq 1+\frac{\left|x-x_0\right|}{|y-x|} \leq
1+\frac{\hat{\rho}_j}{\rho_j-\hat{\rho}_j}\le2^{j+4} .\]Thus, we obtain \[\begin{align}
{\label{J954ex}} J_4&= \theta\mathop{\mathrm{ess\,sup}}_{x \in \text{supp} (\phi_j)} \int_{\mathbb{R}^d\setminus B_{j}} \frac{w_j(y)^{p-1}}{|x-y|^{d+ps}} {\,\rm d}y\cdot \fint_{B_{j}} w_j\phi_j^{p} {\,\rm d}x\nonumber\\ &\le \theta C
2^{j(d+sp)}\int_{\mathbb{R}^d\setminus B_{j}} \frac{w_j(y)^{p-1}}{|x_0-y|^{d+ps}} {\,\rm d}y\cdot \fint_{B_{j}} w_j\phi_j^{p} {\,\rm d}x\nonumber\\ &\le\theta C 2^{j(d+sp)}\left(\int_{\mathbb{R}^d\setminus B_{j}} \frac{k_j^{p-1}}{|x_0-y|^{d+ps}} {\,\rm
d}y+\int_{\mathbb{R}^d\setminus B_{R}} \frac{(u(y)^-)^{p-1}}{|x_0-y|^{d+ps}} {\,\rm d}y\right)\cdot \fint_{B_{j}} w_j{\,\rm d}x\nonumber\\ &\le C 2^{j(d+sp)}\left(k_j^{p-1} r^{-ps}+r^{-p}\theta\left(\frac{r}{R}\right)^p \operatorname{Tail}\left(u^{-} ;
x_0, R\right)^{p-1}\right) \cdot \fint_{B_{j}} w_j{\,\rm d}x\nonumber\\ &\le C 2^{j(d+sp)}k_j^{p-1} r^{-p}\fint_{B_{j}} w_j{\,\rm d}x,
\end{align}\tag{57}\] where \(C=C(d,s,p)\). Here we have used the fact that \(r \in(0,1]\) along with 49 , \(\delta
k<k_j\) and \(u \geq 0\) a.e. in \(B_R\). Also, observe that \[\begin{align}
{\label{sigmap}}\fint_{B_j}w_j^p{\,\rm d}x\leq \left(\fint_{B_j}w_j^{p\sigma}{\,\rm d}x\right)^{\frac{1}{\sigma}}.
\end{align}\tag{58}\] Merging 53 , 55 , 56 and 57 , we obtain from 52 using 58 ,
\[\begin{align}
{\label{expan102}} \fint_{B_{j}} \phi_j^{p} |\nabla w_j|^{p} {\,\rm d}x \le&\, C 2^{jp}r^{-p}\left(\fint_{B_j}w_j^{p\sigma}{\,\rm d}x\right)^{\frac{1}{\sigma}}+C r^{\frac{d}{\sigma}-d}k_j^p\left(\frac{\left|B_j
\cap\left\{u<k_j\right\}\right|}{|B_j|}\right)^{\frac{1}{\sigma}}\nonumber\\&+C2^{j(d+sp)}k_j^{p-1} r^{-p}\fint_{B_{j}} w_j{\,\rm d}x\nonumber\\ \leq &\,C 2^{jp}r^{-p}k_j^p\left(\frac{\left|B_j
\cap\left\{u<k_j\right\}\right|}{|B_j|}\right)^{\frac{1}{\sigma}}+C r^{\frac{d}{\sigma}-d}k_j^p\left(\frac{\left|B_j \cap\left\{u<k_j\right\}\right|}{|B_j|}\right)^{\frac{1}{\sigma}}\nonumber\\&+C2^{j(d+sp)}k_j^{p-1} r^{-p}\left(\fint_{B_{j}}
w_j^\sigma{\,\rm d}x\right)^{\frac{1}{\sigma}}\nonumber\\\leq &\,C 2^{j{(p+sp+d)}}r^{-p}k_j^p\left(\frac{\left|B_j \cap\left\{u<k_j\right\}\right|}{|B_j|}\right)^{\frac{1}{\sigma}}+C r^{\frac{d}{\sigma}-d}k_j^p\left(\frac{\left|B_j
\cap\left\{u<k_j\right\}\right|}{|B_j|}\right)^{\frac{1}{\sigma}}.
\end{align}\tag{59}\] Here we again use the fact that \(w_j\le k_j\) a.e. in \(B_j \cap \{ u<k_j \}\). By applying the Sobolev inequality as in 8 along with 53 , 58 , 59 and the properties of \(\phi_j\), there exists \(C=C(d, p,
s,\|V\|)\) such that \[\begin{align}
{\label{expan18}}
(k_{j}-{k}_{j+1})^{p}\left(\frac{\left|B_{j+1} \cap\left\{u<k_{j+1}\right\}\right|}{|B_{j+1}|}\right)^{\frac{p}{p^*}}& \leq\left(\fint_{B_{j+1}}({w}_j\phi_j)^{p^*}{\,\rm d}x\right)^{\frac{p}{p^*}} \leq C\left(\fint_{B_{j}}({w}_j\phi_j)^{p^*}{\,\rm
d}x\right)^{\frac{p}{p^*}}\nonumber\\
& \leq C r^p \fint_{B_j}|\nabla(w_j \phi_j)|^p {\,\rm d}x\nonumber\\
& \leq C 2^{j(d+p s+p)} \left(1+r^{p-d+\frac{d}{\sigma}}\right)k_j^p \left(\frac{\left|B_j \cap\left\{u<k_j\right\}\right|}{|B_j|}\right)^{\frac{1}{\sigma}} .
\end{align}\tag{60}\] As \(r\in(0,1)\) and \(p-d+\frac{d}{\sigma}\geq 0\), we have from 60 that \[\begin{align}
{\label{expan20}} (k_{j}-{k}_{j+1})^{p}\left(\frac{\left|B_{j+1} \cap\left\{u<k_{j+1}\right\}\right|}{|B_{j+1}|}\right)^{\frac{p}{p^*}}\leq C 2^{j(d+p s+p)}k_j^p \left(\frac{\left|B_j \cap\left\{u<k_j\right\}\right|}{|B_j|}\right)^{\frac{1}{\sigma}}
.
\end{align}\tag{61}\] Let \[Y_j=\left(\frac{\left|B_j \cap\left\{u<k_j\right\}\right|}{|B_j|}\right)^{\frac{1}{p\sigma}}.\] From 61 and 51 there exists
\(C_2=C_2(d, p, s,\|V\|)>1\) such that \[\begin{align}
{\label{expan21}}
&\left(Y_{j+1}\right)^{\frac{p^2\sigma}{p^*}} \leq C_2 2^{j(d+2 p+p s)} Y_j^p \implies Y_{j+1}\leq C_2\hat{C}^j\left(Y_j\right)^{1+\beta},
\end{align}\tag{62}\] where using \(p\sigma<p^*\), we see that \(\beta:=\frac{p^*}{p\sigma}-1>0\) and \(\hat{C}:=2^{\left(\frac{d+2p+ps}{p}\right)\frac{p^*}{p\sigma}}>1.\) We choose \(c_0=C_2\) and \(b=\hat{C}\) in Lemma 5. By 49 , \[\begin{align}
k_0=\frac{3}{2} \delta k \leq 2 \delta k-\frac{1}{2}\theta^{\frac{1}{p-1}}\left(\frac{r}{R}\right)^{\frac{p}{p-1}} \operatorname{Tail}\left(u^{-} ; x_0, R\right)-\varepsilon.
\end{align}\] By 42 , \[\begin{align}
{\label{expan22}}
Y_0 \leq \Bigg(\frac{\left|B_{6 r}\left(x_0\right) \cap\left\{u \leq 2 \delta k-\frac{1}{2}\theta^{\frac{1}{p-1}}\left(\frac{r}{R}\right)^{\frac{p}{p-1}} \operatorname{Tail}\left(u^{-} ; x_0, R\right)-\varepsilon\right\}\right|}{\left|B_{6
r}\left(x_0\right)\right|}\Bigg)^{\frac{1}{p\sigma}} \leq \left(\frac{C_1}{\tau \log \frac{1}{2 \delta}}\right)^{\frac{1}{p\sigma}},
\end{align}\tag{63}\] for \(C_1=C_1(d, p, s)\) and for every \(\delta \in\left(0, \frac{1}{4}\right)\). Using 63 we now choose \(\delta=\delta(d, p, s, \|V\|, \tau)\) as \[\begin{align}
0<\delta:=\frac{1}{4} \exp \left(-\frac{C_1 c_0^{\frac{p\sigma}{\beta}} b^{\frac{p\sigma}{\beta^2}}}{\tau}\right)<\frac{1}{4},
\end{align}\] so that the estimate \(Y_0 \leq c_0^{-\frac{1}{\beta}} b^{-\frac{1}{\beta^2}}\) holds in Lemma 5. Therefore, in 62 , Lemma 5 infers that \(Y_j \rightarrow 0\) as \(j \rightarrow \infty\). Thus we have
\[\begin{align}
\underset{B_{4 r}\left(x_0\right)}{\operatorname{ess} \inf } u \geq \delta k,
\end{align}\] which gives 48 and so 41 holds. This completes the proof. ◻
Proceeding similarly as in the proof of [8], along with an application of Lemma [expan], we obtain the following preliminary version of the weak Harnack inequality, compared to Theorem [weakhar95intro].
Lemma 8. Assume that \(u\) is a weak supersolution to 11 satisfying \(u \geq 0\) in \(B_R(x_0) \subset \Omega\). Let \(0<r\leq1\) be such that \(B_r(x_0) \subset B_{\frac{R}{2}}(x_0)\). Then there exists \(\zeta_3(d,p)>0\) such that for \(\|V\|_{L^q(\Omega)} \le \zeta_3\), the following holds \[\left(\fint_{B_r(x_0)} u^\lambda{\,\rm d}x\right)^{\frac{1}{\lambda}} \leq C \mathop{\mathrm{ess\,inf}}_{B_r(x_0)} u+C\theta^{\frac{1}{p-1}}\left(\frac{r}{R}\right)^{\frac{p}{p-1}} \operatorname{Tail}\left(u^{-} ; x_0, R\right),\] where \(\lambda=\lambda(d, p, s) \in(0,1)\) and \(C=C(d, p, s) \geq 1\).
From the local boundedness and the Tail estimate, we now obtain another weak Harnack inequality. For that, we require the following iteration lemma from [32].
Lemma 9. Let \(0\leq T_0\leq \tilde{t} \leq T_1\) and assume that \(f:[T_0,T_1]\to[0,\infty)\) is a nonnegative bounded function. Suppose that for \(T_0\leq \tilde{t} <\hat{t} \leq T_1\), we have \[\label{itt} f(\tilde{t})\leq A(\hat{t}-\tilde{t})^{-\alpha} +B +\theta f(\hat{t}),\tag{64}\] where \(A,B,\alpha,\theta\) are nonnegative constants and \(\theta<1\). Then there exists \(C=C(\alpha,\theta)\) such that for every \(\rho,R\) and \(T_0\leq\rho<R\leq T_1\), we have \[\label{itt1} f(\rho)\leq C(A(R-\rho)^{-\alpha}+B).\tag{65}\]
Proposition 6. Assume that \(u\) is a weak supersolution to 11 satisfying \(u \geq 0\) in \(B_R(x_0) \subset \Omega\). Let \(0<r\leq1\) be such that \(B_r(x_0) \subset B_{R}(x_0)\). Then there exists \(\zeta_2(d,p)>0\) such that for \(\|V\|_{L^q(\Omega)} \le \zeta_2\), \[\begin{align} \label{mixed-harnack-1} \mathop{\mathrm{ess\,sup}}_{B_{\frac{r}{2}}(x_0)}\,u \le C(d,s,p) \left( \theta^{\frac{1}{p-1}}\Big(\frac{r}{R}\Big)^\frac{p}{p-1}\text{\rm Tail}(u^{-};x_0,R) + \left(\fint_{B_r(x_0)}u^{t}\right)^{\frac{1}{t}} \right), \end{align}\qquad{(2)}\] where \(t \in (0, p\sigma)\) with \(\sigma=\frac{(dp-d+p)q-d}{(dp-d+p)q-dp}\).
Proof. Let \(0<\rho<r\). As \(\theta\in(0,1]\), using ?? of Proposition 5 and 34 , we get \[\begin{align} \mathop{\mathrm{ess\,sup}}_{B_{\frac{\rho}{2}}(x_0)}u & \leq C \delta \theta^{\frac{1}{p-1}}\left( \theta^{\frac{1}{1-p}}\mathop{\mathrm{ess\,sup}}_{B_\rho(x_0)}\,u+\Big(\frac{\rho}{R}\Big)^\frac{p}{p-1}\text{\rm Tail}(u^{-};x_0,R) \right) +C\delta^{\frac{(1-p)p^*}{p(p^*-p\sigma)}}\left(\fint_{B_\rho(x_0)}u^{p\sigma}\right)^{\frac{1}{p\sigma}}, \\ & \le C(d,s,p) \delta \left(\mathop{\mathrm{ess\,sup}}_{B_\rho(x_0)}\,u+\theta^{\frac{1}{p-1}}\Big(\frac{\rho}{R}\Big)^\frac{p}{p-1}\text{\rm Tail}(u^{-};x_0,R) \right) +C\delta^{\frac{(1-p)p^*}{p(p^*-p\sigma)}}\left(\fint_{B_\rho(x_0)}u^{p\sigma}\right)^{\frac{1}{p\sigma}}, \end{align}\] where \(C=C(d,s,p)\). We set \(\rho=(\eta-\eta')r,\) with \(\frac{1}{2}\leq\eta'<\eta\leq1\). We have by a covering argument that \[\begin{align} \mathop{\mathrm{ess\,sup}}_{B_{\eta' r}(x_0)}u &\le C \left( \frac{\delta^{\frac{(1-p)p^*}{p(p^*-p\sigma)}}}{(\eta - \eta')^{\frac{d}{p \sigma}}} \left( \fint_{B_{\eta r}(x_0)} u^{p \sigma} \right)^{\frac{1}{p \sigma}} + \delta \mathop{\mathrm{ess\,sup}}_{B_{\eta r}(x_0)}u + \delta \theta^{\frac{1}{p-1}}\Big(\frac{r}{R}\Big)^\frac{p}{p-1}\text{\rm Tail}(u^{-};x_0,R) \right), \end{align}\] where for \(t \in (0, p \sigma)\), \[\begin{align} \left( \fint_{B_{\eta r}(x_0)} u^{p \sigma} \right)^{\frac{1}{p \sigma}} \le \left( \mathop{\mathrm{ess\,sup}}_{B_{\eta r}(x_0)}u \right)^{\frac{p \sigma -t}{p \sigma}} \left( \fint_{B_{\eta r}(x_0)} u^{t} \right)^{\frac{1}{p \sigma}}. \end{align}\] Choosing \(\delta=\frac{1}{4C(d,p,s)}\) and applying Young’s inequality with the conjugate pair \((\frac{p \sigma}{p \sigma -t}, \frac{p \sigma}{t})\), we get \[\begin{align} \mathop{\mathrm{ess\,sup}}_{B_{\eta'r}(x_0)}u\leq \frac{1}{2}\mathop{\mathrm{ess\,sup}}_{B_{\eta r}(x_0)}u+\frac{C}{(\eta-\eta')^{\frac{d}{t}}}\left(\fint_{B_{\eta r}(x_0)}u^{t}\right)^{\frac{1}{t}} + C\theta^{\frac{1}{p-1}} \Big(\frac{r}{R}\Big)^\frac{p}{p-1}\text{\rm Tail}(u^{-};x_0,R), \end{align}\] for some \(C=C(d,p,s)\). Now applying Lemma 9, with \(f(z) = \underset{B_{zr}(x_0)}{\mathop{\mathrm{ess\,sup}}}\,u, \hat{t} = \eta , \tilde{t} = \eta', \alpha = \frac{d}{t}\), we obtain \[\begin{align} \mathop{\mathrm{ess\,sup}}_{B_{\frac{r}{2}}(x_0)}\,u \le C(d,s,p) \left( \theta^{\frac{1}{p-1}}\Big(\frac{r}{R}\Big)^\frac{p}{p-1}\text{\rm Tail}(u^{-};x_0,R) + \left(\fint_{B_{r}(x_0)}u^{t}\right)^{\frac{1}{t}} \right), \end{align}\] for every \(t \in (0, p\sigma)\) with \(\sigma=\frac{(dp-d+p)q-d}{(dp-d+p)q-dp}\). This is indeed ?? . ◻
Theorem 7. Assume that \(u\) is a weak solution to 11 satisfying \(u \geq 0\) in \(B_R(x_0) \subset \Omega\). Let \(0<r\leq1\) be such that \(B_r(x_0) \subset B_{\frac{R}{2}}(x_0)\). Then there exists \(\zeta(d,p)>0\) such that for \(\|V\|_{L^q(\Omega)} \le \zeta\), \[\mathop{\mathrm{ess\,sup}}_{B_{\frac{r}{2}}(x_0)} u \le C \mathop{\mathrm{ess\,inf}}_{B_r(x_0)} u + C\theta^{\frac{1}{p-1}}\left(\frac{r}{R}\right)^{\frac{p}{p-1}} \text{\rm Tail}_{p-1,sp,p}(u^-; x_0, R),\] where \(C=C(d,p,s)\).
Proof. We take \(\zeta = \min \{\zeta_2, \zeta_3\}>0\). Since \(p\sigma>1\), we choose \(t=\lambda\) in Proposition [prop], and then combine with Lemma [lower], to conclude the proof. ◻
Theorem 8. Assume that \(u\) is a weak supersolution to 11 satisfying \(u \geq 0\) in \(B_R(x_0) \subset \Omega\). Let \(0<r\leq1\) be such that \(B_r(x_0) \subset B_{\frac{R}{2}}(x_0)\). Then there exists \(\hat{\zeta}(m,d,p)>0\) such that for \(\|V\|_{L^q(\Omega)} \le \hat{\zeta}\), \[\left(\fint_{B_{\frac{r}{2}}\left(x_0\right)} u^l {\,\rm d}x\right)^{\frac{1}{l}} \leq C \mathop{\mathrm{ess\,inf}}_{B_r(x_0)} u+C\theta^{\frac{1}{p-1}}\left(\frac{r}{R}\right)^{\frac{p}{p-1}} \text{\rm Tail}_{p-1,sp,p}\left(u^{-} ; x_0, R\right),\] whenever \(0<l<\kappa(p-1)\), with \(\kappa\) as given in 7 . Here \(C=C(d, p, s)\).
Proof. Let \(r \in(0,1), \frac{1}{2}<\tau^{\prime}<\tau \leq \frac{3}{4}\) and \(\phi \in {\mathcal{C}}_c^{\infty}(B_{\tau r}(x_0))\) be such that \(0 \leq \phi \leq 1\) in \(B_{\tau r}(x_0), \phi=1\) in \(B_{\tau^{\prime} r}(x_0)\) and \(|\nabla \phi| \leq \frac{4}{\left(\tau-\tau^{\prime}\right) r}\). For \(t>0\) and \(m\in(1, p)\), we set \[h=u+t \; \text{ and } \; w=(u+t)^{\frac{p-m}{p}} .\] Observe that \[{\label{weakhar1}} \int_{B_r} w^p|\nabla \phi|^p {\,\rm d}x\leq \frac{C(p) r^{-p}}{\left(\tau-\tau^{\prime}\right)^p} \int_{B_{\tau r}\left(x_0\right)} w^p {\,\rm d}x,\tag{66}\] Noting \(r \in(0,1]\) and using the fact that support of \(\nabla\phi\) is a subset of \(B_{\tau r}\), we have \[\begin{align} {\label{weakhar2}} \theta\iint_{B_r \times B_r} \max \{w(x), w(y)\}^p|\phi(x)-\phi(y)|^p \,{\rm d}\mu&\leq \frac{C(p)r^{-p}}{\left(\tau-\tau^{\prime}\right)^p} \iint_{B_{\tau r}(x_0) \times B_{\tau r}(x_0)} \frac{(w^p(x)+ w^p(y))}{|x-y|^{d+ps-p}} {\,\rm d}x\nonumber\\&\leq r^{p-ps}\frac{C(p) r^{-p}}{\left(\tau-\tau^{\prime}\right)^p} \int_{B_{\tau r}\left(x_0\right)} w^p {\,\rm d}x\nonumber\\&\leq \frac{C(p) r^{-p}}{\left(\tau-\tau^{\prime}\right)^p} \int_{B_{\tau r}\left(x_0\right)} w^p {\,\rm d}x, \end{align}\tag{67}\] for some \(C=C(d,p,s)\). Assume that \(\operatorname{Tail}\left(u^{-} ; x_0, R\right)\) is positive. Then for any \(\varepsilon>0\) and \(r \in(0,1)\) choosing \[t=\frac{1}{2}\theta^{\frac{1}{p-1}}\left(\frac{r}{R}\right)^{\frac{p}{p-1}} \operatorname{Tail}\left(u^{-} ; x_0, R\right)+\varepsilon>0,\] and noting that \[{\label{weakhar3}} \underset{z \in \operatorname{supp} \phi}{\operatorname{ess} \sup } \int_{\mathbb{R}^d\backslash B_r} \frac{{\,\rm d}y}{|z-y|^{d+ps}} \leq C(d, p, s) r^{-p},\tag{68}\] we obtain \[\begin{align} {\label{weakhar4}} &\left(\theta\mathop{\mathrm{ess\,sup}}_{z \in \operatorname{supp} \phi} \int_{\mathbb{R}^d\backslash B_r}\frac{{\,\rm d}y}{|z-y|^{d+ps}} +\theta t^{1-p} R^{-p} \operatorname{Tail}(u^{-} ; x_0, R)^{p-1}\right)\int_{B_r} w^p \phi^p {\,\rm d}x\nonumber\\&\leq C(d, p, s) \frac{r^{-p}}{\left(\tau-\tau^{\prime}\right)^p} \int_{B_{\tau r}(x_0)} w^p {\,\rm d}x, \end{align}\tag{69}\] where we use the fact that \(\tau-\tau'\in (0,1)\). If \(\operatorname{Tail}\left(u^{-} ; x_0, R\right)=0\), we can choose \(t=\varepsilon>0\) and again using 68 the estimate in 69 follows. We take \(\hat{\zeta} = \min \{\zeta_1, \zeta_3\}>0\). Now using Sobolev inequality in 8 and the fact that \(\phi = 1\) in \(B_{\tau^{\prime} r}, r \in(0,1)\), we combine Lemma [3463] with the estimates 66 , 67 , and 69 . Consequently, we get \[\begin{align} \left(\fint_{B_{\tau^{\prime} r}(x_0)} h^{\frac{d(p-m)}{d-p}} {\,\rm d}x\right)^{\frac{p}{p^*}} & =\left(\fint_{B_{\tau^{\prime} r}(x_0)} w^{p^*} {\,\rm d}x\right)^{\frac{p}{p^*}} \leq\left(\fint_{B_{\tau r}(x_0)}|w \phi|^{p^*} {\,\rm d}x\right)^{\frac{p}{p^*}} \nonumber\\ & \leq(\tau r)^{p-d} \int_{B_{\tau r}(x_0)}|\nabla(w \phi)|^p {\,\rm d}x\leq \frac{C}{\left(\tau-\tau^{\prime}\right)^p} \fint_{B_{\tau r}(x_0)} w^p {\,\rm d}x, \end{align}\] where \(C=C(d, p, s, m)\). For \(m \in(1, p)\) using the Moser iteration technique as in [33] and [34], we get \[\begin{align} \left(\fint_{B_{\frac{r}{2}}(x_0)} h^l {\,\rm d}x\right)^{\frac{1}{l}} &\leq C\left(\fint_{B_{\frac{3 r}{4}}(x_0)} h^{l^{\prime}} {\,\rm d}x\right)^{\frac{1}{l^{\prime}}},\; \forall\, 0<l^{\prime}<l<\frac{d(p-1)}{d-p}. \end{align}\] Also observe that \[\fint_{B_{\frac{r}{2}}(x_0)} u^l\leq\fint_{B_{\frac{r}{2}}(x_0)} h^l,\; \forall\, l>0.\] For \(\lambda\in(0,1)\) as in Lemma [lower], we take \(l^{\prime}=\lambda\) to obtain \[\begin{align} {\label{weakhar7}} &{\left(\fint_{B_{\frac{r}{2}}(x_0)} u^l {\,\rm d}x\right)^{\frac{1}{l}}} \leq C\left(\fint_{B_{\frac{3 r}{4}}(x_0)} h^{l^{\prime}} {\,\rm d}x\right)^{\frac{1}{l^{\prime}}} \leq C\left(\fint_{B_{\frac{3 r}{4}}(x_0)} u^{l^{\prime}} {\,\rm d}x\right)^{\frac{1}{l^{\prime}}} +Ct \nonumber\\&{\leq C \mathop{\mathrm{ess\,inf}}_{B_r} u+C\theta^{\frac{1}{p-1}}\left(\frac{r}{R}\right)^{\frac{p}{p-1}} \operatorname{Tail}\left(u^{-} ; x_0, R\right)}+Ct,\; \forall \, 0<l<\frac{d(p-1)}{d-p}, \end{align}\tag{70}\] where \(C=C(l^\prime)\). We have neglected the dependence of \(C\) on \(l^\prime\) as \(l^\prime=\lambda(d,p,s)\). For any \(\varepsilon>0\), choosing \[t=\frac{1}{2}\theta^{\frac{1}{p-1}}\left(\frac{r}{R}\right)^{\frac{p}{p-1}} \operatorname{Tail}\left(u^{-} ; x_0, R\right)+\varepsilon,\] in 70 and letting \(\varepsilon \rightarrow 0\), the result follows. ◻
In the following remark, we note the relation between tail terms, average integral, \(\mathop{\mathrm{ess\,sup}}\) and \(\mathop{\mathrm{ess\,inf}}\) of \(u_{\rho}\) and \(u\).
Remark 9. Recall \(\theta\) from Remark 4. For \(\rho>0\), let \(y_0= \rho x_0\). We observe that \[\begin{align} \theta \{\text{\rm Tail}_{p-1,sp,p}\left(u_{\rho}^- ; x_0, R\right)\}^{p-1} & = R^{p} \rho^{d+sp}\rho^{p-sp}\,\int_{\mathbb{R}^d\setminus B_R(x_0)} \frac{|u^-(\rho x)|^{p-1}}{|\rho x-y_0|^{d+sp}}{\,\rm d}x\\ & = 2^{1-p} R^{p} \rho^{d+sp}\rho^{p-sp}\,\int_{\mathbb{R}^d\setminus B_R(x_0)} \frac{||u(\rho x)|-u(\rho x)|^{p-1}}{|\rho x-y_0|^{d+sp}}{\,\rm d}x\\ & = 2^{1-p} R^{p} \rho^{p} \,\int_{\mathbb{R}^d\setminus B_{\rho R}(y_0)} \frac{||u(y)|-u(y)|^{p-1}}{|y-y_0|^{d+sp}} {\,\rm d}y\\ &= R^{p} \rho^{p} \, \int_{\mathbb{R}^d\setminus B_{\rho R}(y_0)} \frac{|u^-(y)|^{p-1}}{|y-y_0|^{d+sp}} {\,\rm d}y= \{\text{\rm Tail}_{p-1,sp,p}\left(u^- ; y_0, \rho R\right)\}^{p-1}. \end{align}\] We also see that \[\begin{align} & \fint_{B_r(x_0)} u_{\rho}^l {\,\rm d}x= \frac{1}{\rho^d|B_r(x_0)|} \int_{B_{\rho r}(y_0)} u^l {\,\rm d}x= \frac{1}{\rho^d|B_r(y_0)|} \int_{B_{\rho r}(y_0)} u^l {\,\rm d}x= \fint_{B_{\rho r}(y_0)} u^l {\,\rm d}x, \\ & \mathop{\mathrm{ess\,sup}}_{B_{r}(x_0)} u_{\rho} = \mathop{\mathrm{ess\,sup}}_{B_{\rho r}(y_0)} u, \text{ and } \mathop{\mathrm{ess\,inf}}_{B_r(x_0)} u_{\rho} = \mathop{\mathrm{ess\,inf}}_{B_{\rho r}(y_0)} u. \end{align}\]
Proof of Theorem [harnack95intro] and Theorem [weakhar95intro]: For \(\tilde{\Omega}\) as given in Remark 4, we choose \(\rho_0 \in (0,1)\) such that \(\@ifstar{\norm}{\norm*}{V_{\rho_0}}_{\tilde{\Omega}}< \min\{\zeta, \hat{\zeta}\}\) and \(R_0 = \rho_0R\). Then the proof follows combining Theorem [harnack], Theorem [weakharnack], and Remark 9. 0◻
Acknowledgments: The research of N.B. is supported by the National Board for Higher Mathematics Postdoctoral Fellowship (0204/16(9)/2024/RD-II/6761). S.D. acknowledges the financial support provided by Indian Institute of Technology Kanpur.