\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2018 (2018), No. 122, pp. 1--14.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu} \thanks{\copyright 2018 Texas State University.} \vspace{8mm}} \begin{document} \title[\hfilneg EJDE-2018/122\hfil Fractional $p$-Laplacian inclusions] {Two solutions for fractional $p$-Laplacian inclusions under nonresonance} \author[A. Iannizzotto, E. M. Rocha, S. Santos \hfil EJDE-2018/122\hfilneg] {Antonio Iannizzotto, Eug\'enio M. Rocha, Sandrina Santos} \address{Antonio Iannizzotto \newline Department of Mathematics and Computer Science, University of Cagliari, Viale L. Merello 92, 09123 Cagliari, Italy} \email{antonio.iannizzotto@unica.it} \address{Eug\'enio M. Rocha \newline CIDMA - Center for Research and Development in Mathematics and Applications, Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal} \email{eugenio@ua.pt} \address{Sandrina Santos \newline CIDMA - Center for Research and Development in Mathematics and Applications, Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal} \email{sras@ua.pt} \thanks{Submitted February 6, 2018. Published June 15, 2018.} \subjclass[2010]{35R11, 34A60, 49J92, 58E05} \keywords{Fractional $p$-Laplacian; differential inclusion; nonsmooth analysis; \hfill\break\indent critical point theory} \begin{abstract} We study a pseudo-differential inclusion driven by the fractional $p$-Laplacian operator and involving a nonsmooth potential, which satisfies nonresonance conditions both at the origin and at infinity. Using variational methods based on nonsmooth critical point theory (Clarke's subdifferential), we establish existence of at least two constant sign solutions (one positive, the other negative), enjoying H\"older regularity \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{definition}[theorem]{Definition} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{remark}[theorem]{Remark} \newtheorem{example}[theorem]{Example} \allowdisplaybreaks \section{Introduction and main result}\label{sec1} In this article we study the problem \begin{equation}\label{p} \begin{gathered} (-\Delta)_p^s u\in\partial j(x,u) \quad \text{in }\Omega \\ u=0 \quad \text{in }\Omega^c. \end{gathered} \end{equation} Here $\Omega\subset\mathbb{R}^N$ ($N>1$) is a bounded domain with a $C^2$ boundary $\partial\Omega$, $p>1$ and $s\in(0,1)$ are real numbers such that $ps0$, see \cite{CS, DPV}). Moreover, $\partial j(x,\cdot)$ denotes the generalized subdifferential (in the sense of Clarke \cite{C}) of a potential $j:\Omega\times\mathbb{R}\to\mathbb{R}$ which is assumed to be measurable in $\Omega$ and locally Lipschitz continuous in $\mathbb{R}$. Thus, problem \eqref{p} can be referred to as a pseudo-differential inclusion in $\Omega$, coupled with a Dirichlet-type condition in $\Omega^c=\mathbb{R}^N\setminus\Omega$ (due to the nonlocal nature of the operator $(-\Delta)_p^s$). Nonsmooth problems driven by linear and nonlinear operators, such as the $p$-Laplacian, have been extensively studied in a variational perspective, since the pioneering work \cite{C1}. The use of variational methods based on nonsmooth critical point theory allows to establish several existence and multiplicity results for problems related to locally Lipschitz potentials, which can be equivalently formulated as either differential inclusions or hemivariational inequalities, see \cite{BM, CIM, I1, IMM, IP, KP, MN, PRS, PRS1, PSS} and the monographs \cite{GP, MP, NP}. The study of nonlocal problems driven by fractional-type operators (both linear and nonlinear) is more recent but rapidly expanding, because of both the important applications of nonlocal diffusion in several disciplines (for instance, in mechanics, game theory, population dynamics, and probability) and to the intrinsic mathematical interest: indeed, fractional operators induce a class of integral equations, exhibiting many common features with partial differential equations. Out of a vast literature, let us mention the results of \cite{AP, BCSS, CS, CS1, GS, IMS, IP1, RS, SV, SV1} (linear case) \cite{BBS, BP, DQ, FP, ILPS, IMS1, IS, J, LL, MPSY, PSY, PP, S} ($p$-case), as well as \cite{C2, DPV, MRS} for a general introduction to fractional operators. Our work stands at the conjunction of these two branches of research. By applying nonsmooth critical point theory, we shall prove the existence of two constant sign, smooth weak solutions for problem \eqref{p}. Precisely, on the nonsmooth potential $j$ we will assume the following: \begin{itemize} \item[(H1)] $j:\Omega\times\mathbb{R}\to\mathbb{R}$ is a function such that $j(\cdot,0)=0$, $j(\cdot,t)$ is measurable in $\Omega$ for all $t\in\mathbb{R}$, $j(x,\cdot)$ is locally Lipschitz continuous in $\mathbb{R}$ for a.e.\ $x\in\Omega$. Moreover, \item[(H2)] for all $\rho>0$ there exists $a_\rho\in L^\infty(\Omega)_+$ such that for a.e.\ $x\in\Omega$, all $|t|\le\rho$, and all $\xi\in\partial j(x,t)$, $|\xi|\le a_\rho(x)$; \item[(H3)] there exists $\theta\in L^\infty(\Omega)_+$ such that $\theta\le\lambda_1$, $\theta\not\equiv\lambda_1$, and uniformly for a.e.\ $x\in\Omega$, \[ \limsup_{|t|\to\infty}\max_{\xi\in\partial j(x,t)}\frac{\xi}{|t|^{p-2}t} \le\theta(x); \] \item[(H4)] there exist $\eta_1,\eta_2\in L^\infty(\Omega)_+$ such that $\lambda_1\le\eta_1\le\eta_2$, $\eta_1\not\equiv\lambda_1$, and uniformly for a.e.\ $x\in\Omega$ \[ \eta_1(x)\le\liminf_{t\to 0}\min_{\xi\in\partial j(x,t)}\frac{\xi}{|t|^{p-2}t} \le\limsup_{t\to 0}\max_{\xi\in\partial j(x,t)}\frac{\xi}{|t|^{p-2}t}\le\eta_2(x); \] \item[(H5)] for a.e.\ $x\in\Omega$, all $t\in\mathbb{R}$, and all $\xi\in\partial j(x,t)$ $\xi t\ge 0$. \end{itemize} In (H3) and (H4), $\lambda_1>0$ denotes the principal eigenvalue of $(-\Delta)_p^s$ with Dirichlet conditions in $\Omega$ (see Section \ref{sec2} below), so these conditions conjure a nonresonance phenomenon both at infinity and at the origin. Here we present an example of a potential satisfying (H1): \begin{example}\label{ex} \rm Let $\theta,\eta\in L^\infty(\Omega)_+$ be such that $\theta<\lambda_1<\eta$, and $j:\Omega\times\mathbb{R}\to\mathbb{R}$ be defined for all $(x,t)\in\Omega\times\mathbb{R}$ by \[ j(x,t)=\begin{cases} \frac{\eta(x)}{p}|t|^p & \text{if } |t|\le 1 \\[4pt] \frac{\theta(x)}{p}|t|^p+\ln(|t|^p)+\frac{\eta(x)-\theta(x)}{p} & \text{if } |t|>1. \end{cases} \] Then $j$ satisfies (H1)--(H5). \end{example} To the best of our knowledge, this is the first existence/multiplicity result for a nonlocal problem involving fractional operators and set-valued reactions in higher dimension, while we should mention \cite{YT, ZG} for the ordinary case (the first based on fixed point methods, the second on nonsmooth variational methods). We also recall a nice application of nonsmooth analysis to a single-valued nonlocal equation in \cite{DS}. Our main result is as follows: \begin{theorem}\label{main} If hypotheses {\rm (H1)--(H5)} hold, then problem \eqref{p} admits at least two solutions $u_+,u_-\in C^\alpha(\overline\Omega)$ ($\alpha\in (0,1)$) such that $u_-(x)<00$ such that \[ |\varphi(v)-\varphi(w)|\leq L\|v-w\| \text{ for all } v,w\in U. \] From now on we assume $\varphi$ to be locally Lipschitz continuous. The \emph{generalized directional derivative} of $\varphi$ at $u$ along $v\in X$ is \[ \varphi^\circ(u;v)=\limsup_{w\to u,\, t\to 0^+} \frac{\varphi(w+tv)-\varphi(w)}{t}. \] The \emph{generalized subdifferential} of $\varphi$ at $u$ is the set \[ \partial\varphi(u)=\big\{u^*\in X^*: \langle u^*,v\rangle\leq\varphi^\circ(u;v) \text{ for all } v\in X\big\}. \] We say that $u$ is a \emph{critical point} of $\varphi$ if $0\in\partial\varphi(u)$. The following Lemmas display some useful properties of the notions introduced above, see \cite[Propositions 1.3.7-1.3.12]{GP}: \begin{lemma}\label{gd} If $\varphi,\,\psi:X\to\mathbb{R}$ are locally Lipschitz continuous, then \begin{itemize} \item[(i)] $\varphi^\circ(u;\cdot)$ is positively homogeneous, sub-additive and continuous for all $u\in X$; \item[(ii)] $\varphi^\circ(u;-v)=(-\varphi)^\circ(u;v)$ for all $u,v\in X$; \item[(iii)] if $\varphi\in C^1(X)$, then $\varphi^\circ(u;v)=\langle\varphi'(u),v\rangle$ for all $u,v\in X$; \item[(iv)] %\label{gd4} $(\varphi+\psi)^\circ(u;v)\leq\varphi^\circ(u;v)+\psi^\circ(u;v)$ for all $u,v\in X$. \end{itemize} \end{lemma} \begin{lemma}\label{gg} If $\varphi,\psi:X\to\mathbb{R}$ are locally Lipschitz continuous, then \begin{itemize} \item[(i)] %\label{gg1} $\partial\varphi(u)$ is convex, closed and weakly$^*$ compact for all $u\in X$; \item[(ii)] %\label{gg2} the multifunction $\partial\varphi:X\to 2^{X^*}$ is upper semicontinuous with respect to the weak$^*$ topology on $X^*$; \item[(iii)] %\label{gg3} if $\varphi\in C^1(X)$, then $\partial\varphi(u)=\{\varphi'(u)\}$ for all $u\in X$; \item[(iv)] %\label{gg4} $\partial(\lambda\varphi)(u)=\lambda\partial\varphi(u)$ for all $\lambda\in\mathbb{R}$, $u\in X$; \item[(v)] %\label{gg5} $\partial(\varphi+\psi)(u)\subseteq\partial\varphi(u)+\partial\psi(u)$ for all $u\in X$; \item[(vi)] %\label{gg6} if $u$ is a local minimizer (or maximizer) of $\varphi$, then $0\in\partial\varphi(u)$. \end{itemize} \end{lemma} Now we deal with integral functionals defined on $L^p$-spaces by means of locally Lipschitz continuous potentials. Let $\Omega\subset\mathbb{R}^N$ be as in the Introduction and $j_0$ be a potential satisfying \begin{itemize} \item[(H6)] $j_0:\Omega\times\mathbb{R}\to\mathbb{R}$ is a function such that $j_0(\cdot,t)$ is measurable in $\Omega$ for all $t\in\mathbb{R}$, $j_0(x,\cdot)$ is locally Lipschitz continuous in $\mathbb{R}$ for a.e.\ $x\in\Omega$. Moreover, there exists $a_0>0$ such that for a.e. $x\in\Omega$, all $t\in\mathbb{R}$, and all $\xi\in\partial j_0(x,t)$ \[|\xi|\le a_0|t|^{p-1}.\] \end{itemize} For $u\in L^p(\Omega)$, we set \[ J_0(u)=\int_\Omega j_0(x,u)\,dx, \] and define the set-valued Nemytzkij operator \[ N_0(u)=\{w\in L^{p'}(\Omega):\,w(x)\in\partial j_0(x,u(x)) \text{ for a.e. } x\in\Omega\}. \] From \cite[Theorem 2.7.5]{C} we have the following statement. \begin{lemma}\label{j0} If $j_0$ satisfies {\rm (H6)}, then $J_0:L^p(\Omega)\to\mathbb{R}$ is Lipschitz continuous on any bounded subset of $L^p(\Omega)$. Moreover, for all $u\in L^p(\Omega)$, $w\in\partial J_0(u)$ one has $w\in N_0(u)$. \end{lemma} Now we collect some useful results related to the fractional $p$-Laplacian defined in \eqref{fpl}. First we fix a functional-analytical framework, following \cite{DPV,ILPS}. For all measurable $u:\mathbb{R}^N\to\mathbb{R}$ we define the Gagliardo seminorm $[u]_{s,p}$ by setting \[ [u]_{s,p}^p=\iint_{\mathbb{R}^N\times\mathbb{R}^N}\frac{|u(x)-u(y)|^p}{|x-y|^{N+ps}}\,dx\,dy, \] then we introduce the fractional Sobolev space \[ W^{s,p}(\mathbb{R}^N)=\big\{u\in L^p(\mathbb{R}^N):\,[u]_{s,p}<\infty\big\}, \] endowed with the norm $\|u\|_{s,p}^p=\|u\|_p^p+[u]_{s,p}^p$. Letting $\Omega$ be as in the Introduction, and taking into account the Dirichlet-type condition, we restrict ourselves to the space \[ W^{s,p}_0(\Omega)=\big\{u\in W^{s,p}(\mathbb{R}^N):\,u(x)=0 \text{ for a.e. } x\in\Omega^c \big\}. \] Because of the fractional Poincar\'e inequality (see \cite[Theorem 7.1]{DPV}), $ W^{s,p}_0(\Omega)$ can be normed by means of $\|u\|=[u]_{s,p}$. With such a norm, $( W^{s,p}_0(\Omega),\|\cdot\|)$ is a separable, uniformly convex (hence, reflexive) Banach space. We set \[ p^*_s=\frac{Np}{N-ps}. \] Then the embedding $ W^{s,p}_0(\Omega)\hookrightarrow L^q(\Omega)$ is continuous for all $q\in[1,p^*_s]$ and compact for all $q\in[1,p^*_s)$ (in particular, for $q=p$), see \cite[Corollary 7.2]{DPV}. Moreover, we denote by $( W^{s,p}_0(\Omega)^*,\|\cdot\|_*)$ the topological dual of $( W^{s,p}_0(\Omega),\|\cdot\|)$. The operator $(-\Delta)_p^s$ can be represented by a duality mapping $A: W^{s,p}_0(\Omega)\to W^{s,p}_0(\Omega)^*$ defined for all $u,v\in W^{s,p}_0(\Omega)$ by \[ \langle A(u),v\rangle = \iint_{\mathbb{R}^N\times\mathbb{R}^N}\frac{|u(x)-u(y)|^{p-2}(u(x) -u(y))(v(x)-v(y))}{|x-y|^{N+ps}}\,dx\,dy, \] which satisfies the (S)-property, namely, whenever $u_n\rightharpoonup u$ in $ W^{s,p}_0(\Omega)$ and \[ \lim_n\langle A(u_n),u_n-u\rangle=0, \] then we have $u_n\to u$ in $ W^{s,p}_0(\Omega)$ (see \cite{ILPS, PAO}). Now we consider the \eqref{p}-type problem \begin{equation}\label{p0} \begin{gathered} (-\Delta)_p^s u\in\partial j_0(x,u) \quad \text{in } \Omega \\ u=0 \quad \text{in } \Omega^c, \end{gathered} \end{equation} where $j_0$ satisfies (H6). We introduce the following notion of weak (or variational) solution. \begin{definition}\label{sol} \rm A function $u\in W^{s,p}_0(\Omega)$ is a (weak) solution of \eqref{p0} if there exists $w\in N_0(u)$ such that for all $v\in W^{s,p}_0(\Omega)$ \[ \langle A(u),v\rangle = \int_\Omega wv\,dx. \] \end{definition} Recalling that $ W^{s,p}_0(\Omega)\hookrightarrow L^p(\Omega)$, conversely we have $L^{p'}(\Omega)\hookrightarrow W^{s,p}_0(\Omega)^*$, so Definition \ref{sol} can be rephrased by \begin{equation}\label{wp0} A(u)=w \quad \text{in } W^{s,p}_0(\Omega)^*. \end{equation} By means of \eqref{wp0}, problem \eqref{p} is somewhat reduced to a pseudodifferential equation (with single-valued right hand side), to which we can apply most recent results from fractional calculus of variations. We begin with uniform $L^\infty$-bounds, whose proof closely follows that of \cite[Theorem 3.1]{ILPS}: \begin{lemma}\label{bound} If $j_0$ satisfies {\rm (H6)}, then there exists $C_0>0$ such that for all solution $u\in W^{s,p}_0(\Omega)$ of \eqref{p0} one has $u\in L^\infty(\Omega)$ and \[ \|u\|_\infty\le C_0(1+\|u\|). \] \end{lemma} \begin{proof} Without loss of generality we may assume $u^+\not\equiv 0$ (the case $u^-\not\equiv 0$ is analogous). By Definition \ref{sol}, there exists $w\in N_0(u)$ such that \eqref{wp0} holds. By (H6) we have for a.e.\ $x\in\Omega$ \begin{equation}\label{bound1} |w(x)|\le a_0|u(x)|^{p-1}. \end{equation} Choose $\rho\ge\max\{1,\|u\|_p^{-1}\}$ (to be determined later) and set $v=(\rho\|u\|_p)^{-1}u\in W^{s,p}_0(\Omega)$, so that $\|v\|_p=\rho^{-1}\le 1$ and $A(v)=(\rho\|u\|_p^{-1})^{p-1}A(u)$ by $(p-1)$-homogeneity of the fractional $p$-Laplacian. For all $n\in\mathbb{N}$ set \[ v_n=\Big(v-1+\frac{1}{2^n}\Big)^+\in W^{s,p}_0(\Omega), \] in particular $v_0=v^+$. The sequence $(v_n)$ is pointwise nonincreasing as for all $n\in\mathbb{N}$, a.e.\ $x\in\Omega$ we have $0\le v_{n+1}(x)\le v_n(x)$, and $v_n(x)\to (v(x)-1)^+$ as $n\to\infty$ for a.e.\ $x\in\Omega$. Moreover we have for all $n\in\mathbb{N}$, \begin{equation}\label{bound2} \big\{v_{n+1}>0\big\} \subseteq \big\{0\frac{1}{2^{n+1}}\big\}. \end{equation} Indeed, for a.e.\ $x\in\Omega$ such that $v_{n+1}(x)>0$ we have $v(x)>1-2^{-n-1}>1-2^{-n}$, hence $v(x)>0$ and $v_n(x)>2^{-n-1}$. Further, we have \begin{align*} (2^{n+1}-1)v_n(x)-v(x) &= (2^{n+1}-2)v(x)+(2^{n+1}-1)\Big(\frac{1}{2^n}-1\Big) \\ &= (2^{n+1}-2)\Big(v(x)-1+\frac{1}{2^{n+1}}\Big) > 0, \end{align*} which proves \eqref{bound2}. Set $R_n=\|v_n\|_p^p$, so $(R_n)$ is a nonincreasing sequence in $(0,1)$. We claim that \begin{equation}\label{bound3} \lim_n R_n=0. \end{equation} Indeed, for all $n\in\mathbb{N}$, we have \begin{align*} R_{n+1} &= \int_{\{v_{n+1}>0\}}v_{n+1}^p\,dx \\ &\le \|v_{n+1}\|_{p^*_s}^p\big|\big\{v_n>\frac{1}{2^{n+1}}\big\} \big|^\frac{p^*_s-p}{p^*_s} \quad \text{(by H\"older's inequality and \eqref{bound2})} \\ &\le \|v_{n+1}\|_{p^*_s}^p\Big(2^{p(n+1)} \int_{\{v_n>2^{-n-1}\}}v_n^p\,dx\Big)^\frac{p^*_s-p}{p^*_s} \quad \text{(by Chebyshev's inequality)} \\ &\le c_1\|v_{n+1}\|^p\Big(2^{p(n+1)}R_n\Big)^\frac{p^*_s-p}{p^*_s} \; \text{(by the fractional Sobolev inequality, $c_1>0$).} \end{align*} Besides, testing \eqref{wp0} with $v_{n+1}\in W^{s,p}_0(\Omega)$ we have \begin{align*} \|v_{n+1}\|^p &\le \langle A(v),v_{n+1}\rangle \\ &= (\rho\|u\|_p)^{1-p}\int_\Omega wv_{n+1}\,dx \\ &\le (\rho\|u\|_p)^{1-p}\int_{\{v_{n+1}>0\}}a_0|u|^{p-1}v_{n+1}\,dx \quad \text{(by \eqref{bound1})} \\ &= a_0\int_{\{v_{n+1}>0\}}|v|^{p-1}v_{n+1}\,dx \\ &\le a_0\int_{\{v_{n+1}>0\}}\big((2^{n+1}-1)v_n)^{p-1}v_{n+1}\,dx \quad \text{(by \eqref{bound2})} \\ &\le a_0(2^{n+1}-1)^{p-1}\int_{\{v_n>0\}}v_n^p\,dx \quad \text{(by monotonicity of $(v_n)$)} \\ &= a_0(2^{n+1}-1)^{p-1}R_n. \end{align*} Concatenating the above inequalities, we obtain the recursive formula \begin{equation}\label{bound4} R_{n+1}\le H^{n+1}R_n^{1+\beta}, \end{equation} where the constants $H>1$, $\beta\in(0,1)$ (independent of $u$) are given by \[ H=\max\{1,a_0c_1\}2^{2p-1}, \ \beta=\frac{p^*_s-p}{p^*_s}. \] Now we fix \[ \rho=\max\big\{1,\|u\|_p^{-1},H^\frac{1+\beta}{p\beta^2}\big\}, \quad \eta=H^{-\frac{1}{\beta}}\in(0,1). \] We have for that all $n\in\mathbb{N}$, \begin{equation}\label{bound5} R_n\le\frac{\eta^n}{\rho^p}. \end{equation} We argue by induction on $n\in\mathbb{N}$. Clearly $R_0\le\rho^{-p}$. If \eqref{bound5} holds for some $n\ge 1$, then by \eqref{bound4} we have \[ R_{n+1}\le H^{n+1}\Big(\frac{\eta^n}{\rho^p}\Big)^{1+\beta} \le \frac{\eta^{n+1}}{\rho^p}. \] Recalling that $\eta\in(0,1)$, from \eqref{bound5} we deduce \eqref{bound3}. Thus we have $v_n\to 0$ in $L^p(\Omega)$. Passing if necessary to a subsequence we have $v_n(x)\to 0$ for a.e.\ $x\in\Omega$, which, along with $v_n(x)\to(v(x)-1)^+$, implies $v(x)\le 1$ for a.e.\ $x\in\Omega$. An analogous argument applies to $-v$, therefore we have $v\in L^\infty(\Omega)$ with $\|v\|_\infty\le 1$. Going back to $u$ and recalling the definition of $\rho$, we have $u\in L^\infty(\Omega)$ with \begin{align*} \|u\|_\infty &\le \rho\|u\|_p \\ &= \max\{\|u\|_p,1,H^\frac{1}{p\beta^2}\|u\|_p\} \\ &\le C_0(1+\|u\|) \end{align*} for some $C_0>0$ which does not depend on $u$. \end{proof} \begin{remark} \rm If in (H6) the exponent $p-1$ is replaced by $q-1$ for some $q\in(1,p^*_s)$, some uniform $L^\infty$-bounds still hold (see \cite{ILPS}). We kept this special assumption for the sake of simplicity. \end{remark} Weak solutions exhibit H\"older regularity up to the boundary. For $\alpha\in(0,1)$, we shall use the function space $C^\alpha(\overline\Omega)$, endowed with the norm \[ \|u\|_{C^\alpha(\overline\Omega)}=\|u\|_\infty +\sup_{x,y\in\Omega,\,x\neq y}\frac{|u(x)-u(y)|}{|x-y|^\alpha}. \] \begin{lemma}\label{hr} If $j_0$ satisfies {\rm (H6)}, then there exist $\alpha\in(0,s]$, $K_0>0$ such that for all solution $u\in W^{s,p}_0(\Omega)$ of \eqref{p0} one has $u\in C^\alpha(\overline\Omega)$ and \[ \|u\|_{C^\alpha(\overline\Omega)}\le K_0(1+\|u\|). \] \end{lemma} \begin{proof} By Lemma \ref{bound} we have $u\in L^\infty(\Omega)$ and $\|u\|_\infty\le C_0(1+\|u\|)$, with $C_0>0$ independent of $u$. Let $w\in N_0(u)$ be as in Definition \ref{sol}, then by (H6) we have \[ \|w\|_\infty\le a_0\|u\|_\infty^{p-1}\le c_2(1+\|u\|^{p-1}) \] for some $c_2>0$ independent of $u$. Now \cite[Theorem 1.1]{IMS1} implies $u\in C^\alpha(\overline\Omega)$ and \[ \|u\|_{C^\alpha(\overline\Omega)}\le c_3\|w\|_\infty^\frac{1}{p-1} \le K_0(1+\|u\|), \] with $c_3,K_0>0$ independent of $u$. \end{proof} No regularity higher than $C^s$ can be expected in the fractional framework, as was pointed out in \cite{RS} even for the linear case (fractional Laplacian). In particular, solutions do not, in general, admit an outward normal derivative at the points of $\partial\Omega$ and, as a consequence, the Hopf property is stated in terms of a H\"older-type quotient (see \cite{DQ} and Lemmas \ref{varp} and \ref{varm} below). Similarly to the case of the $p$-Laplacian ($s=1$), the spectrum of $(-\Delta)_p^s$ includes a sequence $0<\lambda_1<\lambda_2\le\ldots\le\lambda_k\le\ldots$ of variational eigenvalues with min-max characterizations (see \cite{BP, FP, IS, LL, PSY, S} for a detailed description of such eigenvalues). Here we shall only use the following properties of $\lambda_1$: \begin{lemma}\label{l1} The principal eigenvalue $\lambda_1$ of $(-\Delta)_p^s$ in $ W^{s,p}_0(\Omega)$ is simple and isolated (as en element of the spectrum), with the following variational characterization: \[ \lambda_1=\inf_{u\in W^{s,p}_0(\Omega)\setminus\{0\}}\frac{\|u\|^p}{\|u\|_p^p}. \] The corresponding positive, $L^p(\Omega)$-normalized eigenfunction is $u_1\in C^\alpha(\overline\Omega)$. \end{lemma} The following result is crucial in obtaining the constant sign solutions of \eqref{p}, exploiting hypothesis (H1) (H2) (nonresonance at infinity): \begin{proposition}\label{coer} Let $\theta\in L^\infty(\Omega)_+$ be such that $\theta\le\lambda_1$, $\theta\not\equiv\lambda_1$, and $\psi\in C^1( W^{s,p}_0(\Omega))$ be defined by \[ \psi(u)=\|u\|^p-\int_\Omega\theta(x)|u|^p\,dx. \] Then there exists $\theta_0\in(0,\infty)$ such that for all $u\in W^{s,p}_0(\Omega)$ \[ \psi(u)\ge\theta_0\|u\|^p. \] \end{proposition} \begin{proof} By Lemma \ref{l1} we have for all $u\in W^{s,p}_0(\Omega)$ \[ \psi(u)\ge\|u\|^p-\lambda_1\|u\|_p^p\ge 0. \] To complete the proof, we argue by contradiction: we assume that there exists a sequence $(u_n)$ in $ W^{s,p}_0(\Omega)$ such that $\|u_n\|=1$ for all $n\in\mathbb{N}$, and $\psi(u_n)\to 0$. By reflexivity of $ W^{s,p}_0(\Omega)$ and the compact embedding $ W^{s,p}_0(\Omega)\hookrightarrow L^p(\Omega)$, passing if necessary to a subsequence we have $u_n\rightharpoonup u$ in $ W^{s,p}_0(\Omega)$ and $u_n\to u$ in $L^p(\Omega)$, as well as $u_n(x)\to u(x)$ for a.e.\ $x\in\Omega$. By convexity \[ \liminf_n\|u_n\|^p\ge\|u\|^p, \] while by the dominated convergence theorem \[ \lim_n\int_\Omega\theta(x)|u_n|^p\,dx=\int_\Omega\theta(x)|u|^p\,dx. \] Thus we obtain $\psi(u)=0$. Two cases may occur: \begin{itemize} \item[(a)] if $u=0$, then we obtain \[ \|u_n\|^p=\psi(u_n)+\int_\Omega\theta(x)|u_n|^p\,dx \to 0, \] against $\|u_n\|=1$; \item[(b)] if $u\neq 0$, then $u$ is a minimizer of the Rayleigh quotient in Lemma \ref{l1}, hence by simplicity of the principal eigenvalue we can find $\mu\in\mathbb{R}$ such that $u=\mu u_1$, in particular $|u(x)|>0$ for all $x\in\Omega$, which in turn implies \[ \|u\|^p=\int_\Omega\theta(x)|u|^p\,dx<\lambda_1\|u\|_p^p, \] against the variational characterization of $\lambda_1$. \end{itemize} So we have \[ \inf_{\|u\|=1}\psi(u)=\theta_0>0, \] and noting that $\psi$ is $p$-homogeneous we complete the proof. \end{proof} \section{Proof of the main result}\label{sec3} In this section we prove Theorem \ref{main}. First we establish a variational framework for problem \eqref{p} introducing two truncated, nonsmooth energy functionals. For all $(x,t)\in\Omega\times\mathbb{R}$ set \[ j_\pm(x,t)=j(x,\pm t^\pm), \] and for all $u\in W^{s,p}_0(\Omega)$ set \[ \varphi_\pm(u)=\frac{\|u\|^p}{p}-\int_\Omega j_\pm(x,u)\,dx. \] The functionals $\varphi_\pm$ select constant sign solutions of \eqref{p}, as explained by the following lemmas: \begin{lemma}\label{varp} The functional $\varphi_+: W^{s,p}_0(\Omega)\to\mathbb{R}$ is locally Lipschitz continuous. Moreover, if $u\in W^{s,p}_0(\Omega)\setminus\{0\}$ is a critical point of $\varphi_+$, then $u\in C^\alpha(\overline\Omega)$ is a solution of \eqref{p} such that \begin{itemize} \item[(i)] %\label{varp1} $u(x)>0$ for all $x\in\Omega$; \item[(ii)] % \label{varp2} for all $y\in\partial\Omega$ \[ \liminf_{\Omega\ni x\to y}\frac{u(x)}{{\rm dist}(x,\Omega^c)^s}>0. \] \end{itemize} \end{lemma} \begin{proof} First we note that $j_+(\cdot,t)$ is measurable in $\Omega$ for all $t\in\mathbb{R}$ and $j_+(x,\cdot)$ is locally Lipschitz continuous in $\mathbb{R}$ for a.e.\ $x\in\Omega$, with generalized subdifferential $\partial j_+(x,\cdot)$ such that for all $t\in\mathbb{R}$, \begin{equation}\label{jp1} \partial j_+(x,t)\begin{cases} =\{0\} & \text{if } t<0 \\ \subseteq\{\mu\xi:\,\mu\in[0,1],\,\xi\in\partial j(x,0)\} & \text{if } t=0 \\ =\partial j(x,t) & \text{if } t>0. \end{cases} \end{equation} Moreover, there exists $c_4>0$ such that for a.e.\ $x\in\Omega$, all $t\in\mathbb{R}$, and all $\xi\in\partial j_+(x,t)$ \begin{equation}\label{jp2} |\xi|\le c_4|t|^{p-1}. \end{equation} Indeed, by \eqref{jp1}, the inequality above holds if $t<0$. Now fix $\varepsilon>0$. By (H3) (H5) we can find $\rho>0$ such that for a.e.\ $x\in\Omega$, all $t>\rho$, and all $\xi\in\partial j(x,t)$ we have \[ 0\le\xi\le (\|\theta\|_\infty+\varepsilon)t^{p-1}, \] while by (H4) (H5) we can find $\delta\in(0,\rho)$ such that for a.e.\ $x\in\Omega$, all $00$ for all $x\in\Omega$, which proves (i). By \eqref{jp1}, the latter estimate implies $w(x)\in\partial j(x,u(x))$ for a.e.\ $x\in\Omega$, hence by Definition \ref{sol} $u$ solves \eqref{p}. Finally, by the Hopf lemma for the fractional $p$-Laplacian \cite[Theorem 1.5]{DQ}, we have for all $y\in\partial\Omega$ \[ \liminf_{\Omega\ni x\to y}\frac{u(x)}{{\rm dist}(x,\Omega^c)^s}>0, \] which yields (ii) and completes the proof. \end{proof} An analogous argument leads to the following result. \begin{lemma}\label{varm} The functional $\varphi_-: W^{s,p}_0(\Omega)\to\mathbb{R}$ is locally Lipschitz continuous. Moreover, if $u\in W^{s,p}_0(\Omega)\setminus\{0\}$ is a critical point of $\varphi_-$, then $u\in C^\alpha(\overline\Omega)$ is a solution of \eqref{p} such that \begin{itemize} \item[(i)] %\label{varm1} $u(x)<0$ for all $x\in\Omega$; \item[(ii)] % \label{varm2} for all $y\in\partial\Omega$ \[ \limsup_{\Omega\ni x\to y}\frac{u(x)}{{\rm dist}(x,\Omega^c)^s}<0. \] \end{itemize} \end{lemma} We can now prove our main result. \begin{proof}[Proof of Theorem \ref{main}] We deal first with the positive solution, which is detected as a global minimizer of the truncated functional $\varphi_+$. By (H1) (H3), for any $\varepsilon>0$ we can find $\rho>0$ such that for a.e.\ $x\in\Omega$, all $t>\rho$ and all $\xi\in\partial j_+(x,t)$ \[ |\xi|\le(\theta(x)+\varepsilon)t^{p-1} \] (recall that $\partial j_+(x,t)=\partial j(x,t)$ by \eqref{jp1}). By (H2) and \eqref{jp1} again, there exists $a_\rho\in L^\infty(\Omega)_+$ such that for a.e.\ $x\in\Omega$, all $t\le\rho$ and all $\xi\in\partial j_+(x,t)$ \[ |\xi|\le a_\rho(x). \] So, for a.e.\ $x\in\Omega$, all $t\in\mathbb{R}$ and all $\xi\in\partial j_+(x,t)$ we have \begin{equation}\label{m1} |\xi|\le a_\rho(x)+(\theta(x)+\varepsilon)|t|^{p-1}. \end{equation} By the Rademacher theorem and \cite[Proposition 2.2.2]{C}, for a.e.\ $x\in\Omega$ the mapping $j_+(x,\cdot)$ is differentiable for a.e.\ $t\in\mathbb{R}$ with \[ \frac{d}{dt}\,j_+(x,t)\in\partial j_+(x,t). \] So, integrating and using \eqref{m1} we obtain for a.e.\ $x\in\Omega$ and all $t\in\mathbb{R}$ \begin{equation}\label{m2} j_+(x,t)\le a_\rho(x)|t|+(\theta(x)+\varepsilon)\frac{|t|^p}{p}. \end{equation} For all $u\in W^{s,p}_0(\Omega)$, we have \begin{align*} \varphi_+(u) &\ge \frac{\|u\|^p}{p}-\int_\Omega\Big(a_\rho(x)|u|+(\theta(x)+\varepsilon)\frac{|u|^p}{p} \Big)\,dx \quad \text{(by \eqref{m2})} \\ &\ge \frac{1}{p}\Big(\|u\|^p-\int_\Omega\theta(x)|u|^p\,dx\Big) -\|a_\rho\|_\infty\|u\|_1-\frac{\varepsilon}{p}\|u\|_p^p \\ &\ge \frac{1}{p}\Big(\theta_0-\frac{\varepsilon}{\lambda_1}\Big)\|u\|^p-c_5\|u\| \quad (\theta_0,c_5>0), \end{align*} where in the final passage we have used Lemmas \ref{l1}, \ref{coer}, and the continuous embedding $ W^{s,p}_0(\Omega)\hookrightarrow L^1(\Omega)$. If we choose $\varepsilon\in (0,\theta_0\lambda_1)$, the latter tends to $\infty$ as $\|u\|\to\infty$, hence $\varphi_+$ is coercive in $ W^{s,p}_0(\Omega)$. Moreover, the functional $u\mapsto\|u\|^p/p$ is convex, hence weakly lower semi continuous in $ W^{s,p}_0(\Omega)$, while $J_+$ is continuous in $L^p(\Omega)$, which, by the compact embedding $ W^{s,p}_0(\Omega)\hookrightarrow L^p(\Omega)$ and the Eberlein-Smulyan theorem, implies that $J_+$ is sequentially weakly continuous in $ W^{s,p}_0(\Omega)$. So, $\varphi_+$ is sequentially weakly l.s.c.\ in $ W^{s,p}_0(\Omega)$. As a consequence, there exists $u_+\in W^{s,p}_0(\Omega)$ such that \begin{equation}\label{m3} \varphi_+(u_+)=\inf_{u\in W^{s,p}_0(\Omega)}\varphi_+(u)=m_+. \end{equation} By Lemma \ref{gg} (vi), $u_+$ is a critical point of $\varphi_+$. We claim now that \begin{equation}\label{m4} m_+<0. \end{equation} Indeed, by (H4), for any $\varepsilon>0$ we can find $\delta>0$ such that for a.e.\ $x\in\Omega$, all $t\in[0,\delta)$, and all $\xi\in\partial j_+(x,t)$ \[ \xi\ge(\eta_1(x)-\varepsilon)t^{p-1}. \] As above, integrating we have \begin{equation}\label{m5} j_+(x,t)\ge\frac{\eta_1(x)-\varepsilon}{p}t^p. \end{equation} Let $u_1\in W^{s,p}_0(\Omega)\cap C^\alpha(\overline\Omega)$ be defined as in Lemma \ref{l1}. We can find $\mu>0$ such that $0<\mu u_1(x)\le\delta$ for all $x\in\Omega$. Then we have \begin{align*} \varphi_+(\mu u_1) &\le \frac{\mu^p}{p}\|u_1\|^p-\frac{\mu^p}{p}\int_\Omega(\eta_1(x)-\varepsilon)u_1^p\,dx \quad \text{(by \eqref{m5})} \\ &= \frac{\mu^p}{p}\Big(\int_\Omega(\lambda_1-\eta_1(x))u_1^p\,dx+\varepsilon\Big) \quad \text{(by Lemma \ref{l1}).} \end{align*} Recalling that $\eta_1\ge\lambda_1$ with $\eta_1(x)>\lambda_1$ for all $x$ in a subset of $\Omega$ with positive measure, and that $u_1(x)>0$ for all $x\in\Omega$, we see that \[ \int_\Omega(\lambda_1-\eta_1(x))u_1^p\,dx<0. \] So, for $\varepsilon>0$ small enough, the estimates above imply $\varphi_+(\mu u_1)<0$. Thus, we have \eqref{m4}. In particular, from \eqref{m3} we have $u_+\neq 0$. Now Lemma \ref{varp} implies that $u_+\in C^\alpha(\overline\Omega)$, $u_+(x)>0$ for all $x\in\Omega$, for all $y\in\partial\Omega$ \[ \liminf_{\Omega\ni x\to y}\frac{u_+(x)}{{\rm dist}\,(x,\Omega^c)}>0, \] and finally that $u_+$ is a solution of \eqref{p}. An analogous argument, applied to $\varphi_-$ with the support of Lemma \ref{varm}, proves the existence of another solution $u_-\in C^\alpha(\overline\Omega)$ such that $u_-(x)<0$ for all $x\in\Omega$, \[ \limsup_{\Omega\ni x\to y}\frac{u_-(x)}{{\rm dist}\,(x,\Omega^c)}<0 \] for all $y\in\partial\Omega$. So the proof is concluded. \end{proof} \subsection*{Acknowledgements} This work was partially supported by FCT (Funda\c{c}\~ao para a Ci\^encia e a Tecnologia) trough CIDMA (Center for Research and Development in Mathematics and Applications), within Project UID/MAT/04106/2013. It was partially accomplished during a visit of E. M. 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