\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2018 (2018), No. 127, pp. 1--21.\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-2018/127\hfil Asymmetric Robin BV problems] {Asymmetric Robin boundary-value problems with $p$-Laplacian and indefinite potential} \author[S. A. Marano, N. S. Papageorgiou \hfil EJDE-2018/127\hfilneg] {Salvatore A. Marano, Nikolaos S. Papageorgiou} \address{Salvatore A. Marano (corresponding author)\newline Department of Mathematics and Computer Sciences, University of Catania, Viale A. Doria 6, 95125 Catania, Italy} \email{marano@dmi.unict.it} \address{Nikolaos S. Papageorgiou \newline Department of Mathematics, National Technical University of Athens, Zografou Campus, Athens 15780, Greece} \email{npapg@math.ntua.gr} \dedicatory{Communicated by Vicentiu D. Radulescu} \thanks{Submitted June 2, 2018. Published June 18, 2018.} \subjclass[2010]{35J20, 35J60, 58E05} \keywords{Robin boundary condition; $p$-Laplacian; indefinite potential; \hfill\break\indent asymmetric reaction; superlinear at $+\infty$; resonance; multiple solutions} \begin{abstract} Four nontrivial smooth solutions to a Robin boundary-value problem with $p$-Laplacian, indefinite potential, and asymmetric nonlinearity super-linear at $+\infty$ are obtained, all with sign information. The semilinear case is also investigated, producing a nonzero fifth solution. Our proofs use variational methods, truncation techniques, and Morse theory. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{remark}[theorem]{Remark} \allowdisplaybreaks \section{Introduction} Let $\Omega$ be a bounded domain in $\mathbb{R}^n$ with a $C^2$-boundary $\partial\Omega$, let $a\in L^\infty(\Omega)$, and let $f:\Omega\times\mathbb{R}\to\mathbb{R}$ be a Carath\'eodory function such that $f(\cdot,0)=0$. Consider the Robin problem \begin{equation}\label{prob} \begin{gathered} -\Delta_p u+a(x)|u|^{p-2}u=f(x,u) \quad \text{in }\Omega,\\ \frac{\partial u}{\partial n_p}+\beta(x) |u|^{p-2}u=0 \quad \text{on } \partial\Omega, \end{gathered} \end{equation} where $1< p<+\infty$, $\Delta_p$ indicates the $p$-Laplacian, $\frac{\partial u}{\partial n_p}:=|\nabla u|^{p-2}\nabla u\cdot n$, with $n$ being the outward unit normal vector to $\partial\Omega$, and $\beta\in C^{0,\alpha}(\partial\Omega,\mathbb{R}^+_0)$. We say that $u\in W^{1,p}(\Omega)$ is a (weak) solution of \eqref{prob} provided $$ \int_\Omega|\nabla u|^{p-2}\nabla u\cdot\nabla v\, dx +\int_{\partial\Omega}\beta|u|^{p-2} uv\, d\sigma +\int_\Omega a|u|^{p-2}uv\, dx =\int_\Omega f(x,u)v\, dx $$ for all $ v\in W^{1,p}(\Omega)$. This paper studies the existence of multiple solutions to \eqref{prob} when \begin{itemize} \item the potential function $x\mapsto a(x)$ is indefinite, i.e., sign changing, and \item the reaction term $(x,t)\mapsto f(x,t)$ exhibits an asymmetric behaviour as $t$ goes from $-\infty$ to $+\infty$. \end{itemize} For $(x,\xi)\in\Omega\times\mathbb{R}$, we define \begin{equation}\label{defFH} F(x,\xi):=\int_0^\xi f(x,\tau)d\tau,\quad H(x,\xi):=f(x,\xi)\xi-pF(x,\xi)\,. \end{equation} Roughly speaking, our assumptions on the rate of $f$ at infinity are the following. \begin{itemize} \item[(1)] $ \lim_{\xi\to+\infty}F(x,\xi)\xi^{-p}=+\infty$ uniformly in $x\in\Omega$ and there exists $c_1>0$ such that $$ H(x,\xi_1)\leq H(x,\xi_2)+c_1\quad\text{whenever}\quad 0\leq\xi_1\leq\xi_2. $$ \item[(2)] For appropriate $c_2\in\mathbb{R}$ one has $$ c_2\leq\liminf_{t\to-\infty}\frac{f(x,t)}{|t|^{p-2}t} \leq \limsup_{t\to-\infty}\frac{f(x,t)}{|t|^{p-2}t}\leq \hat\lambda_1\,, \quad \lim_{\xi\to-\infty}H(x,\xi)=+\infty $$ uniformly in $x\in\Omega$. \end{itemize} Here $\hat\lambda_n$ denotes the $n^{\rm th}$-eigenvalue of the problem \begin{equation}\label{eigen} -\Delta_p u+a(x)|u|^{p-2}u=\lambda |u|^{p-2}u\quad\text{in}\quad\Omega, \quad\frac{\partial u}{\partial n_p}+\beta(x) |u|^{p-2}u=0\quad\text{on } \partial\Omega\,. \end{equation} It should be noted that a possible interaction (resonance) with $\hat\lambda_1$ is allowed and that $f(x,\cdot)$ grows $(p-1)$-super-linearly near $+\infty$. Nevertheless, contrary to most previous works, we do not need here the stronger unilateral Ambrosetti-Rabinowitz condition. Under (1), (2), and some additional hypotheses, one of which forces a $p$-concave behaviour of $t\mapsto f(x,t)$ at zero, there are four $C^1$-solutions to \eqref{prob}, two positive, one negative, and the remaining nodal; see Section 3. If $p:=2$ then \eqref{prob} becomes \begin{equation}\label{probp=2} \begin{gathered} -\Delta u+a(x)u=f(x,u) \quad \text{in }\Omega,\\ \frac{\partial u}{\partial n}+\beta(x) u=0 \quad \text{on } \partial\Omega. \end{gathered} \end{equation} As in \cite{DMP,MaPaJMAA2016}, the assumptions on $a$ and $\beta$ can be significantly relaxed. However, we obtain five nontrivial smooth solutions; cf. Theorem \ref{fivesol}. The adopted approach exploits variational methods, truncation techniques, and results from Morse theory. Regularity is a standard matter, unless $p:=2$, in which case \cite[Lemmas 5.1, 5.2]{Wa} are employed. Problem \eqref{probp=2} has been widely investigated under various points of view; see, for instance, \cite{DMP,MaPaJMAA2016} and the references given there. On the contrary, the equation $$ -\Delta_p u+a(x)|u|^{p-2}u=f(x,u) \quad\text{in}\quad\Omega\,, $$ with Dirichlet, Neumann, or Robin boundary conditions, did not receive much attention when $p\neq 2$, a sign-changing potential appears, and $t\mapsto f(x,t)$ is asymmetric. Actually, we can only mention \cite{MoMoPaPRSE}, where the Dirichlet problem is studied, \cite{MuPa}, dealing with symmetric reactions and Neumann boundary conditions, \cite{APT,FMP}, devoted to $(p-1)$-super-linear reactions. The situation looks somewhat different if $a\equiv 0$; vide, e.g., \cite{FiPa,MaPaMFM,PaRa1,PaWi} and their bibliographies. \section{Preliminaries} Let $(X,\|\cdot\|)$ be a real Banach space. Given a set $V\subseteq X$, write $\overline{V}$ for the closure of $V$, $\partial V$ for the boundary of $V$, and $\operatorname{int}_X(V)$ or simply $\operatorname{int}(V)$, when no confusion can arise, for the interior of $V$. If $x\in X$ and $\delta>0$ then $$ B_\delta(x):=\{ z\in X:\| z-x\|<\delta\}\,. $$ The symbol $(X^*,\|\cdot \|_{X^*})$ denotes the dual space of $X$, $\langle\cdot,\cdot\rangle$ indicates the duality pairing between $X$ and $X^*$, while $x_n\to x$ (respectively, $x_n\rightharpoonup x$) in $X$ means `the sequence $\{x_n\}$ converges strongly (respectively, weakly) in $X$'. We say that $\Phi:X\to\mathbb{R}$ is coercive if $$ \lim_{\| x\|\to+\infty}\Phi(x)=+\infty. $$ A function $\Phi$ is called weakly sequentially lower semi-continuous when $$ x_n\rightharpoonup x\quad\text{in } X \implies \Phi(x)\leq\liminf_{n\to\infty}\Phi(x_n). $$ Let $\Phi\in C^1(X)$. The classical Cerami compactness condition for $\Phi$ reads as follows. \begin{itemize} \item[(C)] Every sequence $\{x_n\}\subseteq X$ such that $\{\Phi(x_n)\}$ is bounded and $$ \lim_{n\to+\infty}(1+\| x_n\|)\| \Phi'(x_n) \|_{X^*}=0 $$ has a convergent subsequence. \end{itemize} For $c\in\mathbb{R}$, we define $$ \Phi^c:=\{x\in X: \Phi(x)\leq c\}\,,\quad K_c(\Phi):=K(\Phi)\cap\Phi^{-1}(c)\,, $$ where, as usual, $K(\Phi)$ denotes the critical set of $\Phi$, i.e., \[ K(\Phi):=\{x\in X:\,\Phi'(x)=0\}. \] We say that $A:X\to X^*$ is of type $({\rm S})_+$ if $$ x_n\rightharpoonup x\quad \text{ in } X,\quad \limsup_{n\to+\infty}\langle A(x_n),x_n-x\rangle\leq 0 \implies x_n\to x. $$ Given a topological pair $(A,B)$ fulfilling $B\subset A\subseteq X$, the symbol $H_q(A,B)$, $q\in\mathbb{N}_0$, indicates the ${\rm q}^{\rm th}$-relative singular homology group of $(A,B)$ with integer coefficients. If $x_0\in K_c(\Phi)$ is an isolated point of $K(\Phi)$ then $$ C_q(\Phi,x_0):=H_q(\Phi^c\cap V,\Phi^c\cap V\setminus\{x_0\})\,,\quad q\in \mathbb{N}_0\,, $$ are the critical groups of $\Phi$ at $x_0$. Here, $V$ stands for any neighborhood of $x_0$ such that $K(\Phi)\cap\Phi^c\cap V=\{x_0\}$. By excision, this definition does not depend on the choice of $V$. Suppose $\Phi$ satisfies condition (C), $\Phi\lfloor_{K(\Phi)}$ is bounded below, and $c< \inf_{x\in K(\Phi)}\Phi(x)$. Put $$ C_q(\Phi,\infty):=H_q(X,\Phi^c)\,,\quad q\in\mathbb{N}_0\,. $$ The second deformation lemma \cite[Theorem 5.1.33]{GaPaNA} implies that this definition does not depend on the choice of $c$. If $K(\Phi)$ is finite, then setting \[ M(t,x):=\sum_{q=0}^{+\infty}\operatorname{rank} C_q(\Phi,x)t^q\,,\quad P(t,\infty):=\sum_{q=0}^{+\infty}\operatorname{rank} C_q(\Phi,\infty)t^q\quad \] for $(t,x)\in\mathbb{R}\times K(\Phi)$, the following Morse relation holds \begin{equation}\label{morse} \sum_{x\in K(\Phi)}M(t,x)=P(t,\infty)+(1+t)Q(t)\,, \end{equation} where $Q(t)$ denotes a formal series with nonnegative integer coefficients; see for instance \cite[Theorem 6.62]{MoMoPa}. Now, let $X$ be a Hilbert space, let $x\in K(\Phi)$, and let $\Phi$ be $C^2$ in a neighborhood of $x$. If $\Phi''(x)$ turns out to be invertible, then $x$ is called non-degenerate. The Morse index $d$ of $x$ is the supremum of the dimensions of the vector subspaces of $X$ on which $\Phi''(x)$ turns out to be negative definite. When $x$ is non-degenerate and with Morse index $d$ one has \begin{equation}\label{kd} C_q(\Phi,x)=\delta_{q,d}\mathbb{Z}\,,\quad q\in\mathbb{N}_0\,. \end{equation} The monograph \cite{MoMoPa} represents a general reference on the subject. Throughout this article, $\Omega$ denotes a bounded domain of the real Euclidean $N$-space $(\mathbb{R}^N,|\cdot|)$ whose boundary $\partial\Omega$ is $C^2$ while $n(x)$ indicates the outward unit normal vector to $\partial\Omega$ at its point $x$. On $\partial\Omega$ we will employ the $(N-1)$-dimensional Hausdorff measure $\sigma$. The symbol $m$ stands for the Lebesgue measure, $p\in (1,+\infty)$, $p':=p/(p-1)$, $\|\cdot\|_q$ with $q\geq 1$ is the usual norm of $L^q(\Omega)$, $X:=W^{1,p}(\Omega)$, and $$ \| u\|:=\left(\|\nabla u\|_p^p+\| u\|_p^p\right)^{1/p},\quad u\in X. $$ Write $p^*$ for the critical exponent of the Sobolev embedding $W^{1,p}(\Omega)\subseteq L^q(\Omega)$. Recall that $p^*=Np/(N-p)$ if $p0\; \forall x\in\Omega,\quad \frac{\partial u}{\partial n}\big|_{\partial\Omega\cap u^{-1}(0)}<0 \text{ if }\partial\Omega\cap u^{-1}(0)\neq\emptyset\big\}\,, $$ one evidently has $D_+=\{u\in C_+:u(x)>0\;\forall x\in\overline{\Omega}\}$ as well as \[ D_+\subseteq\hat C_+\subseteq \operatorname{int}(C_+)\,. \] Let $A_p:X\to X^*$ be the nonlinear operator stemming from the negative $p$-Laplacian $\Delta_p$, i.e., \[ \langle A_p(u),v\rangle:=\int_\Omega\vert\nabla u(x)\vert^{p-2} \nabla u(x)\cdot\nabla v(x)\, dx\quad\forall u,v\in X. \] A standard argument \cite[Proposition 2.72]{MoMoPa} ensures that $A_p$ is of type $({\rm S})_+$. \begin{remark} \rm Given $u\in X$, $w\in L^{p'}(\Omega)$, and $\beta\in C^{0,\alpha}(\partial\Omega,\mathbb{R}^+_0)$, the assertion $$ \langle A_p(u),v\rangle+\int_{\partial\Omega}\beta(x)|u(x)|^{p-2}u(x)v(x)d\sigma =\int_\Omega w(x)v(x)dx,\quad v\in X, $$ is equivalent to $$ -\Delta_p u=w\text{ in}\quad\Omega,\quad \frac{\partial u}{\partial n_p}+\beta(x)|u|^{p-2}u=0\text{ on}\quad \partial\Omega. $$ This easily stems from the nonlinear Green's identity \cite[Theorem 2.4.54]{GaPaNA}; see for instance the proof of \cite[Proposition 3]{PaRa}. \end{remark} We shall employ some facts about the spectrum of the operator $$ u\mapsto-\Delta_p u+a(x)|u|^{p-2}u $$ in $X$ with homogeneous Robin boundary conditions. So, consider the eigenvalue problem \eqref{eigen}, where, henceforth, \begin{equation}\label{abeta} a\in L^\infty(\Omega)\text{ and }\beta\in C^{0,\alpha}(\partial\Omega,\mathbb{R}^+_0) \text{ with }\alpha\in (0,1)\,. \end{equation} Define \begin{equation}\label{defE} \mathcal{E} (u):=\|\nabla u\|_p^p+\int_\Omega a(x)|u(x)|^pdx +\int_{\partial\Omega}\beta(x)|u(x)|^pd\sigma\quad\forall u\in X. \end{equation} The Liusternik-Schnirelman theory provides a strictly increasing sequence $\{\hat\lambda_n\}$ of eigenvalues for \eqref{eigen}. Denote by $E(\hat\lambda_n)$ the eigenspace corresponding to $\hat\lambda_n$. As in \cite{MuPa,PaRa}, one has \begin{gather} \label{p1} \text{$\hat\lambda_1$ is isolated and simple. Further, $\hat\lambda_1=\inf_{u\in X\setminus\{0\}}\frac{\mathcal{E}(u)}{\| u\|_p^p}$.} \\ \label{p2} \text{There exists an $L^p$-normalized eigenfunction $\hat u_1\in D_+$ associated with $\hat\lambda_1$.} \end{gather} Let $p:=2$. It is known \cite{DMP,MaPaJMAA2016} that $H^1(\Omega)=\overline{\oplus_{n=1}^\infty E(\hat\lambda_n)}$ and that, for any $n\geq 2$, \begin{equation} \label{p3} \hat\lambda_n=\inf\big\{\frac{\mathcal{E}(u)}{\| u\|_2^2}:u\in\hat H_n,\, u\neq 0\big\} =\sup\big\{\frac{\mathcal{E}(u)}{\| u\|_2^2}:u\in\bar H_n,\, u\neq 0\big\}\,, \end{equation} where $$ \bar H_m:=\oplus_{n=1}^m E(\hat\lambda_n),\quad \hat H_m:=\oplus_{n=m}^\infty E(\hat\lambda_n)\,. $$ \section{Existence results} To avoid unnecessary technicalities, for every $x\in\Omega$' will take the place of `for almost every $x\in\Omega$' while $c_1,c_2, \ldots$ indicate positive constants arising from the context. Henceforth, $f:\Omega\times\mathbb{R}\to\mathbb{R}$ denotes a Carath\'eodory function such that $f(\cdot,0)=0$. Let $F$ and $H$ be given by \eqref{defFH}. We shall make the following assumptions. \begin{itemize} \item[(A1)] There exist $a_1\in L^\infty(\Omega)$ and $r\in (p,p^*)$ such that $$ |f(x,t)|\leq a_1(x)(1+|t|^{r-1})\quad\forall (x,t)\in\Omega\times\mathbb{R}. $$ \item[(A2)] $ \lim_{\xi\to+\infty} F(x,\xi) \xi^{-p}=+\infty$ uniformly in $x\in\Omega$. Moreover, for appropriate $a_2\in L^1(\Omega)_+$, \begin{equation}\label{propH} 0\leq\xi_1\leq\xi_2 \implies H(x,\xi_1)\leq H(x,\xi_2)+a_2(x)\quad\forall x\in\Omega. \end{equation} \item[(A3)] There exists $\bar u\in D_+$ fulfilling $$ \frac{\partial\bar u}{\partial n}\Big|_{\partial\Omega}<0,\quad \Delta_p\bar u\in L^{p'}(\Omega),\quad \langle A_p(\bar u),v\rangle\geq 0\quad\forall v\in W^{1,p}(\Omega)_+\,, $$ and $ \operatorname{ess\,sup}_{x\in\Omega}[f(x,\bar u(x))-a(x)\bar u(x)^{p-1}]<0$. \item[(A4)] For some $a_3\in L^\infty(\Omega)$ one has $$ a_3(x)\leq\liminf_{t\to-\infty}\frac{f(x,t)}{|t|^{p-2}t} \leq\limsup_{t\to-\infty}\frac{f(x,t)}{|t|^{p-2}t}\leq\hat\lambda_1, \quad \lim_{\xi\to-\infty}H(x,\xi)=+\infty $$ uniformly with respect to $x\in\Omega$. \item[(A5)] There exist $q\in (1,p)$ and $\delta_1>0$ satisfying % $$ 00$. \item[(A6)] To every $\rho>0$ there corresponds $\mu_\rho>0$ such that $t\mapsto f(x,t)+\mu_\rho t^{p-1}$ is nondecreasing on $[0,\rho]$ for all $x\in\Omega$. \end{itemize} \begin{remark} \rm The assumption $ \lim_{\xi\to+\infty}F(x,\xi)\xi^{-p}=+\infty$ is weaker than the unilateral Ambrosetti-Rabinowitz condition below. \begin{itemize} \item[(AR)] For appropriate $\theta>p$ and $M>0$ one has $ \operatorname{ess\,inf}_{x\in\Omega} F(x,M)>0$ and $$ 0<\theta F(x,\xi)\leq f(x,\xi)\xi\quad\text{in }\Omega\times [M,+\infty). $$ \end{itemize} A standard example is $f(x,t):=t^{p-1}\log t$, $t\geq M>1$. \end{remark} \begin{remark} \rm Property \eqref{propH} has been thoroughly investigated in \cite[Lemma 2.4]{LY}. Among other things, this result ensures that (A2) forces $ \lim_{t\to+\infty}f(x,t)t^{-p+1}=+\infty$, i.e., $f(x,\cdot)$ turns out to be $(p-1)$-super-linear at $+\infty$. \end{remark} \begin{remark}\label{fthree} \rm Assumption (A3) implies $\Delta_p\bar u\leq 0$. Indeed, via the nonlinear Green's identity \cite[Theorem 2.4.54]{GaPaNA} we get $$ \int_\Omega v(x)\, \Delta_p\bar u(x)\, dx =-\langle A_p(\bar u),v\rangle+\langle\frac{\partial\bar u}{\partial n_p}, v\rangle_{\partial\Omega}\leq 0\quad\forall v\in W^{1,p}(\Omega)_+\,. $$ Here, $\langle\cdot,\cdot\rangle_{\partial\Omega}$ denotes the duality pairing between $W^{-\frac{1}{p'},p'}(\partial\Omega)$ and $W^{\frac{1}{p'},p}(\partial\Omega)$. Moreover, $$ \langle A_p(\overline{u}),v\rangle+\int_\Omega a(x)\overline{u}(x)^{p-1}v(x)\,dx \geq\int_\Omega f(x,\overline{u}(x))v(x)\, dx,\quad v\in W^{1,p}(\Omega)_+, $$ whence $\overline{u}$ is a super-solution of \eqref{prob}. \end{remark} \begin{remark}\label{ffour} \rm Reasoning as in \cite[Lemma 3.1]{DMP} shows that (A4) entails $$ \lim_{\xi\to-\infty}[\hat\lambda_1|\xi|^p-pF(x,\xi)] =+\infty\quad\text{uniformly with respect to}\; x\in\Omega\,. $$ Problem \eqref{prob} is thus coercive in the negative direction, and direct methods can be used to find a negative solution. \end{remark} \begin{remark} \rm After integration, (A5) easily leads to \begin{equation}\label{cone} \theta|\xi|^q\leq F(x,\xi)\quad\forall (x,\xi)\in\Omega\times[-\delta_1,\delta_1], \end{equation} with suitable $\theta>0$. Consequently, $f(x,\cdot)$ exhibits a concave behaviour at zero. \end{remark} We start by pointing out some auxiliary results. \begin{proposition}\label{auxtwo} Suppose $0\leq a$. If $h_i\in L^\infty(\Omega)$, $u_i\in C^1(\overline{\Omega})$, $i=1,2$, fulfill \begin{itemize} \item $-\Delta_p u_i+a(x)|u_i|^{p-2}u_i=h_i $ in $\Omega$, \item $\operatorname{ess\,inf}_{x\in K}[ h_2(x)-h_1(x)]>0$ for any compact set $K\subseteq\Omega$, \item $u_1\leq u_2$ and $\frac{\partial u_2}{\partial n}<0$ on $\partial\Omega$, \end{itemize} then $u_2-u_1\in\hat C_+$. \end{proposition} \begin{proof} Recall that $a\in L^\infty(\Omega)$. The first conclusion, namely $u_2(x)-u_1(x)>0$ for all $x\in\Omega$, is achieved arguing exactly as in the proof of \cite[Proposition 2.6]{ArRu}, while the other directly follows from \cite[Theorem 5.5.1]{PuSe}. \end{proof} \begin{proposition}\label{regularity} Let {\rm (A3)} and {\rm (A6)} be satisfied. Then each nontrivial solution $\tilde u\in[0,\bar u]$ to \eqref{prob} lies in $\operatorname{int} (C_+)\cap(\bar u-\hat C_+)$. \end{proposition} \begin{proof} Standard regularity arguments ensure that $\tilde u\in C_+\setminus\{0\}$. Fix $$ \rho:=\|\bar u\|_\infty\geq\|\tilde u\|_\infty>0. $$ Assumption (A6) provides $\mu_\rho>\| a\|_\infty$ fulfilling $$ -\Delta_p \tilde u(x)+(a(x)+\mu_\rho) \tilde u(x)^{p-1} =f(x,\tilde u(x))+\mu_\rho\tilde u(x)^{p-1}\geq 0\quad\text{a.e. in}\quad\Omega\,. $$ Therefore, by \cite[Theorem 5]{Va}, $\tilde u\in\hat C_+\subseteq \operatorname{int}(C_+)$. Next, define $u_\delta:=\tilde u+\delta$, where $\delta>0$. Since \begin{align*} -\Delta_p \tilde u+(a+\mu_\rho)\tilde u^{p-1} &\leq -\Delta_p\tilde u+(a+\mu_\rho) u_\delta^{p-1}\\ &=-\Delta_p \tilde u+(a+\mu_\rho) \tilde u^{p-1}+o(\delta)\\ &=f(x,\tilde u)+\mu_\rho \tilde u^{p-1}+o(\delta), \end{align*} using (A6) and (A3), with appropriate $c_1>0$, we obtain \begin{align*} -\Delta_p\tilde u+(a+\mu_\rho)\tilde u^{p-1} &\leq f(x,\bar u)+\mu_\rho\bar u^{p-1}+o(\delta) \\ &\leq (a+\mu_\rho)\bar u^{p-1}-c_1+o(\delta) \\ &\leq(a+\mu_\rho)\bar u^{p-1}-\frac{c_1}{2} \\ &\leq-\Delta_p\bar u+(a+\mu_\rho)\bar u^{p-1}-\frac{c_1}{2}, \end{align*} for any $\delta>0$ small enough, because $\Delta_p\bar u\leq 0$; cf. Remark \ref{fthree}. Proposition \ref{auxtwo} now gives $\bar u-\tilde u\in\hat C_+$, as desired. \end{proof} To simplify notation, write $X:=W^{1,p}(\Omega)$. The energy functional $\varphi:X\to\mathbb{R}$ stemming from problem \eqref{prob} is \begin{equation}\label{defphi} \varphi(u):=\frac{1}{p}\mathcal{E}(u)-\int_\Omega F(x,u(x))\, dx,\quad u\in X, \end{equation} with $\mathcal{E}$ and $F$ given by \eqref{defE} and \eqref{defFH}, respectively. One clearly has $\varphi\in C^1(X)$. \begin{proposition}\label{cerami} Under \eqref{abeta}, {\rm (A1), (A2)}, and {\rm (A4)}, the functional $\varphi$ satisfies condition {\rm (C)}. \end{proposition} The proof is rather technical but standard (see, e.g., \cite[Proposition 3.2]{MaPaJMAA2016}). So, we omit it. Henceforth \textit{ $\bar a$ will denote a real constant strictly greater than $\| a\|_\infty$.} \subsection{Positive solutions}\label{sect3.1} Truncation-perturbation techniques and minimization methods produce a first positive solution whenever (A3) is assumed. \begin{theorem}\label{firstsol} Let \eqref{abeta}, {\rm (A1), (A3), (A5)}, and {\rm (A6)} be fulfilled. Then \eqref{prob} has a positive solution $u_0\in \operatorname{int}_{C^1(\overline{\Omega})}([0,\bar u])$. Moreover, $u_0$ turns out to be a local minimizer of $\varphi$. \end{theorem} \begin{proof} For $x\in\Omega$ and $t,\xi\in\mathbb{R}$, we define \begin{equation}\label{truncation} \begin{gathered} \bar f(x,t):= \begin{cases} f(x,t^+)+\bar a (t^+)^{p-1} & \text{if }t^+\leq \bar u(x),\\ f(x,\bar u(x))+\bar a\bar u(x)^{p-1} & \text{otherwise}, \end{cases} \\ \bar F(x,\xi):=\int_0^\xi \bar f(x,t)\, dt. \end{gathered} \end{equation} It is evident that the corresponding functional \[\label{phiplus} \bar\varphi(u):=\frac{1}{p}\left(\mathcal{E}(u)+\bar a\| u\|_p^p\right) -\int_\Omega \bar F(x,u(x))\, dx,\quad u\in X, \] belongs to $C^1(X)$. A standard argument, which exploits Sobolev's embedding theorem besides the compactness of the trace operator, ensures that $\bar\varphi$ is weakly sequentially lower semi-continuous. Since, by \eqref{abeta}, the choice of $\bar a$, and \eqref{truncation}, it is coercive, we have \begin{equation}\label{defuzero} \inf_{u\in X}\bar\varphi(u)=\bar\varphi(u_0) \end{equation} for some $u_0\in X$. Set $\delta:=\min\{\delta_1,\min_{x\in\overline{\Omega}}\bar u(x)\}$, where $\delta_1$ is as in (A5). If $\tau\in (0,1)$ complies with $\tau\hat u_1\leq\delta$, then \[ \bar\varphi(\tau\hat u_1)\leq\frac{\tau^p}{p}\mathcal{E}(\hat u_1) -\theta\tau^q\| \hat u_1\|^q_q =\tau^q\Big(\frac{\tau^{p-q}}{p}\hat\lambda_1-\theta\|\hat u_1\|_q^q\Big) \] thanks to \eqref{truncation}, \eqref{cone}, and \eqref{p2}. Thus, for $\tau$ small enough, $\bar\varphi(\tau\hat u_1)<0$, which entails \[ \bar\varphi(u_0)<0=\bar\varphi(0). \] Consequently, $u_0\neq 0$. Through \eqref{defuzero} we get $\bar\varphi'(u_0)=0$, namely \begin{equation}\label{fbar} \langle A_p(u_0),v\rangle+\int_\Omega(a+\bar a)|u_0|^{p-2}u_0v\, dx +\int_{\partial\Omega}\beta |u_0|^{p-2}u_0v\, d\sigma = \int_\Omega\bar f(x,u_0)v\, dx, \end{equation} for $v\in X$. Using \eqref{truncation} and \eqref{fbar} written for $v:=-u_0^-$ produces $$ \min\{1,\bar a-\| a\|_\infty\}\,\| u_0^-\|^p\leq \mathcal{E}(u_0^-) +\bar a\| u_0^-\|_p^p=0, $$ whence $u_0\geq 0$. Now, choose $v:=(u_0-\bar u)^+$ in \eqref{fbar} and observe that \begin{align*} &\int_\Omega\bar f(x,u_0)(u_0-\bar u)^+ dx \\ &=\int_\Omega[f(x,\bar u)+\bar a\bar u^{p-1}](u_0-\bar u)^+ dx\\ &\leq\int_\Omega(a+\bar a)\bar u^{p-1}(u_0-\bar u)^+ dx +\int_{\partial\Omega}\beta u_0^{p-1}(u_0-\bar u)^+ d\sigma \end{align*} because of \eqref{truncation}, (A3), and \eqref{abeta}. This yields \[ \langle A_p(u_0)-A_p(\bar u),(u_0-\bar u)^+\rangle +(\bar a-\| a\|_\infty)\int_\Omega (u_0^{p-1}-\bar u^{p-1})(u_0-\bar u)^+dx\leq 0, \] i.e., $u_0\leq\bar u$. Therefore, both $u_0\in[0,\bar u]\setminus\{0\}$ and $u_0$ solves problem \eqref{prob}, so that, due to Proposition \ref{regularity}, $u_0\in\operatorname{int}(C_+)\cap(\bar u-\hat C_+)$, which implies $u_0\in \operatorname{int}_{C^1(\overline{\Omega})}([0,\bar u])$. Finally, since $$ \varphi\lfloor_{[0,\bar u]}=\bar\varphi\lfloor_{[0,\bar u]}, $$ Equation \eqref{defuzero}, combined with \cite[Proposition 3]{PaRa}, ensures that $u_0$ is a local minimizer for $\varphi$. \end{proof} Critical point arguments produce a second positive solution. \begin{theorem}\label{secondsol} If \eqref{abeta}, {\rm (A1)--(A3), (A5)--(A6)} hold, then \eqref{prob} possesses a solution $u_1\in \operatorname{int}(C_+)\setminus\{ u_0\}$ such that $u_0\leq u_1$. \end{theorem} \begin{proof} For $x\in\Omega$ and $t,\xi\in\mathbb{R}$, we define \begin{equation}\label{trunczero} \begin{gathered} f_0(x,t):=\begin{cases} f(x,u_0(x))+\bar a u_0(x)^{p-1} & \text{if }t\leq u_0(x),\\ f(x,t)+\bar a t^{p-1} & \text{otherwise}, \end{cases} \\ F_0(x,\xi):=\int_0^\xi f_0(x,t)\, dt. \end{gathered} \end{equation} It is evident that the corresponding truncated functional \begin{equation} \label{phizero} \varphi_0(u):=\frac{1}{p}\left(\mathcal{E}(u)+\bar a\| u\|_p^p\right) -\int_\Omega F_0(x,u(x))\, dx,\quad u\in X, \end{equation} belongs to $C^1(X)$ also. A standard argument, which exploits Sobolev's embedding theorem and the compactness of the trace operator, ensures that $\varphi_0$ is weakly sequentially lower semi-continuous.\ \subsection*{Claim 1:} $\varphi_0$ satisfies condition (C). Let $\{u_n\}$ be a sequence in $X$ be such that \begin{gather}\label{boundzero} |\varphi_0(u_n)|\leq c_1\quad\forall n\in\mathbb{N}, \\ \label{limitzero} \lim_{n\to+\infty}(1+\| u_n\|)\|\varphi'_0(u_n)\|_{X^*}=0. \end{gather} Through \eqref{limitzero} one has \begin{equation}\label{Czero} \begin{aligned} &\Big|\langle A_p(u_n),w\rangle+\int_{\partial\Omega}\beta |u_n|^{p-2}u_nw\, d\sigma\\ &+\int_\Omega(a+\bar a)|u_n|^{p-2}u_nw\, dx -\int_\Omega f_0(x,u_n)w\, dx\Big| \\ &\leq\frac{\varepsilon_n\| w\|}{1+\| u_n\|} \quad \forall w\in X, \end{aligned} \end{equation} where $\varepsilon_n\to 0^+$. We first show that $\{u_n\}$ is bounded. This evidently happens once the same holds for both $\{ u_n^-\}$ and $\{ u_n^+\}$. By \eqref{trunczero}, choosing $w:=-u_n^-$ in \eqref{Czero} easily yields $$ \mathcal{E}(u_n^-)+\bar a\| u_n^-\|_p^p\leq c_2. $$ From \eqref{abeta} and the choice of $\bar a$ it thus follows $\| u_n^-\|\leq c_3$. As $n$ was arbitrary, the sequence $\{ u_n^-\}$ turns out to be bounded. So, in particular, on account of \eqref{boundzero}, $$ \mathcal{E}(u_n^+)+\bar a\| u_n^+\|_p^p-p\int_\Omega F_0(x, u_n^+(x))\, dx \leq c_4\quad\forall n\in\mathbb{N}. $$ Since $$ \int_\Omega F_0(x,u_n^+)\, dx=\int_\Omega[F_0(x,u_n^+)-F_0(x,u_0)]dx +\int_\Omega[f(x,u_0)+\bar a u_0^{p-1}]u_0\, dx, $$ an easy computation shows that \begin{equation}\label{in} \mathcal{E}(u_n^+)-p\int_\Omega F(x,u_n^+(x))\, dx\leq c_5,\quad n\in\mathbb{N}. \end{equation} Now, \eqref{Czero} written with $w:=u_n^+$ furnishes \begin{align*} &-\mathcal{E}(u_n^+)-\bar a\| u_n^+\|_p^p+\int_{\Omega_1}[f(x,u_0) +\bar a u_0^{p-1}]u_n^+dx+\int_{\Omega_2}[f(x,u_n^+) +\bar a (u_n^+)^{p-1}]u_n^+dx\\ & \leq\varepsilon_n, \end{align*} where $\Omega_1:=\{x\in\Omega: 0\leq u_n(x)\leq u_0(x)\}$ and $\Omega_2:=\{x\in\Omega: u_n(x)>u_0(x)\}$. Hence, \begin{equation}\label{ine} -\mathcal{E}(u_n^+)+\int_\Omega f(x,u_n^+)u_n^+dx\leq c_6. \end{equation} Inequalities \eqref{in}--\eqref{ine} lead to $$ \int_\Omega H(x,u_n^+(x))\, dx\leq c_7\quad\forall n\in\mathbb{N}. $$ Via the same arguments used in the proof (Claim 1) of \cite[Proposition 3.2]{MaPaJMAA2016}, with $2$ replaced by $p$, we achieve $\| u_n^+\|\leq c_8$. Therefore, $\{u_n\}\subseteq X$ is bounded. As before, and along a subsequence when necessary, one has $u_n\to u$ in $X$. \subsection*{Claim 2:} $K(\varphi_0)\subseteq\{u\in X:\, u_0\leq u\}$. If $u\in K(\varphi_0)$ then \[ \langle A_p(u),v\rangle+\int_\Omega(a+\bar a)|u|^{p-2}uv\, dx+\int_{\partial\Omega}\beta |u|^{p-2}uv\, d\sigma= \int_\Omega f_0(x,u)v\, dx, \] for all $v\in X$. Letting $v:=(u_0-u)^+$ and recalling that $u_0$ solves \eqref{prob} yields \begin{align*} &\langle A_p(u_0)-A_p(u), (u_0-u)^+\rangle +\int_\Omega(a+\bar a)(u_0^{p-1}-|u|^{p-2}u)(u_0-u)^+dx\\ &+\int_{\partial\Omega}\beta (u_0^{p-1}-|u|^{p-2}u)(u_0-u)^+d\sigma=0. \end{align*} By \eqref{abeta} this entails $$ \langle A_p(u_0)-A_p(u), (u_0-u)^+\rangle +\int_\Omega(a+\bar a)(u_0^{p-1}-|u|^{p-2}u)(u_0-u)^+dx\leq 0, $$ whence $u_0\leq u$, because $\bar a>\| a\|_\infty$. We may evidently assume \begin{equation}\label{kphizero} K(\varphi_0)\cap [0,\bar u]=\{u_0\}, \end{equation} otherwise, thanks to Claim 2, there would exist $u_1\in K(\varphi_0)\cap [u_0,\bar u]\setminus\{u_0\}$, i.e., a second solution of \eqref{prob}. Moreover, Proposition \ref{regularity} would give $u_1\in\operatorname{int}(C_+)\cap (\bar u-\hat C_+)$, and the conclusion follows. For every $x\in\Omega$, $t,\xi\in\mathbb{R}$, we put \begin{equation}\label{barfzero} \bar f_0(x,t):= \begin{cases} f_0(x,t) & \text{if }t\leq\bar u(x),\\ f_0(x,\bar u(x)) & \text{otherwise}, \end{cases} \quad \bar F_0(x,\xi):=\int_0^\xi\bar f_0(x,t)\, dt. \end{equation} The associated truncated functional \[ \bar\varphi_0(u):=\frac{1}{p}\left(\mathcal{E}(u)+\bar a\| u\|_p^p\right) -\int_\Omega\bar F_0(x,u(x))\, dx,\quad u\in X, \] belongs to $C^1(X)$ and is coercive. A standard argument, based on the Sobolev embedding theorem and the compactness of the trace operator, ensures that $\bar\varphi_0$ is weakly sequentially lower semi-continuous. So, \begin{equation}\label{baruzero} \inf_{u\in X}\bar\varphi_0(u)=\bar\varphi_0(\bar u_0) \end{equation} for some $\bar u_0\in X$. Since, like in the proof of Theorem \ref{firstsol}, one has $K(\bar\varphi_0)\subseteq[u_0,\bar u]$, \eqref{kphizero}--\eqref{baruzero} produce $\bar u_0=u_0$. Observe now that $$ \bar\varphi_0\lfloor_{[0,\bar u]}=\varphi_0\lfloor_{[0,\bar u]} $$ while, by Theorem \ref{firstsol}, $u_0\in \operatorname{int}_{C^1(\overline{\Omega})}([0,\bar u])$. Thus, due to \cite[Proposition 3]{PaRa}, $u_0$ is a local minimizer for $\varphi_0$. Without loss of generality, suppose $u_0$ isolated in $K(\varphi_0)$, or else \eqref{prob} would possess infinitely many solutions bigger that $u_0$; cf. Claim 2 and \eqref{trunczero}. The same reasoning made in the proof of \cite[Proposition 29]{APS2} provides here $\rho>0$ fulfilling $$ \varphi_0(u_0)<\inf_{u\in\partial B_\rho(u_0)}\varphi_0(u). $$ From \eqref{trunczero} and (A2) it easily follows that $$ \lim_{\tau\to+\infty}\varphi_0(\tau\hat u_1)=-\infty. $$ Claim 1 guarantees that condition (C) holds for $\varphi_0$. Hence, the mountain-pass theorem gives a point $u_1\in K(\varphi_0)\setminus\{u_0\}$. Obviously, $u_0\leq u_1$ by Claim 2 and $u_1$ solves \eqref{prob}. Through the regularity arguments used above we then achieve $u_1\in C^1(\overline{\Omega})$. It remains to check that $u_1\in\operatorname{int}(C_+)$, which can be performed arguing as in the proof of Proposition \ref{regularity}. \end{proof} \subsection{Negative solutions}\label{sect3.2} The minimization method yields a negative solution whenever (A4) is assumed. \begin{theorem} Let \eqref{abeta}, {\rm (A1), (A4)}, and {\rm (A5)} be satisfied. Then \eqref{prob} possesses a solution $u_2\in-\operatorname{int}(C_+)$. \end{theorem} \begin{proof} For $x\in\Omega$ and $t,\xi\in\mathbb{R}$, we define \[ \tilde f(x,t):= \begin{cases} f(x,t)+\bar a |t|^{p-2}t & \text{if }t\leq 0,\\ 0 & \text{otherwise}, \end{cases} \quad \tilde F(x,\xi):=\int_0^\xi\tilde f(x,t)\, dt. \] It is evident that the corresponding functional % \[\label{tildephi} \tilde\varphi(u):=\frac{1}{p}\left(\mathcal{E}(u)+\bar a\| u\|_p^p\right) -\int_\Omega \tilde F(x,u(x))\, dx,\quad u\in X, \] belongs to $C^1(X)$. A standard reasoning, which exploits Sobolev's embedding theorem besides the compactness of the trace operator, ensures that $\tilde\varphi$ turns out to be weakly sequentially lower semi-continuous. Moreover, $\tilde\varphi$ is coercive. Indeed, if \begin{equation}\label{coercivity} \| u_n\|\to+\infty\quad\text{and}\quad \tilde\varphi(u_n)\leq c_1\quad \forall n\in\mathbb{N}, \end{equation} then \begin{equation}\label{boundminus} \begin{aligned} &\frac{1}{p}\mathcal{E}(u_n^-)-\int_\Omega F(x,-u_n^-(x))\, dx \\ &\leq\frac{1}{p}\min\{1,\bar a-\| a\|_\infty\}\| u_n^+\|^p +\frac{1}{p}\mathcal{E}(u_n^-) -\int_\Omega F(x,-u_n^-(x))\, dx \\ &\leq\frac{1}{p}\left(\mathcal{E}(u_n)+\bar a\| u_n\|^p_p\right) -\int_\Omega\tilde F(x,-u_n^-(x))\, dx\leq c_1\,,\quad n\in\mathbb{N}. \end{aligned} \end{equation} Suppose $\| u_n^-\|\to+\infty$ and write $w_n:=\| u_n^-\|^{-1}u_n^-$. From $\| w_n\|=1$ it follows, along a subsequence when necessary, \begin{equation}\label{weakminus} w_n\rightharpoonup w\text{ in }X,\quad w_n\to w\text{ in } L^p(\Omega)\text{ and in } L^p(\partial\Omega),\quad w\geq 0. \end{equation} Through \eqref{boundminus} one has \begin{equation}\label{wn} \frac{1}{p}\mathcal{E}(w_n)-\frac{1}{\| u_n^-\|^p}\int_\Omega F(x,-u_n^-(x))\, dx \leq\frac{c_1}{\| u_n^-\|^p}\quad\forall n\in\mathbb{N} \end{equation} while by (A1) the sequence $\{\| u_n^-\|^{-p}N_F(-u_n^-)\}\subseteq L^1(\Omega)$ is uniformly integrable. Using the arguments made in the proof of \cite[Proposition 14]{APS2}, besides (A4), we thus obtain a function $\theta\in L^\infty(\Omega)$ such that $-c_2\leq\theta\leq\hat \lambda_1/p$ and \begin{equation}\label{deftheta} \frac{1}{\| u_n^-\|^p}N_F(-u_n^-)\rightharpoonup\frac{1}{p}\theta w^p\quad \text{in } L^1(\Omega). \end{equation} Thanks to \eqref{weakminus}--\eqref{wn} this implies, as $n\to+\infty$, \begin{equation}\label{ineqE} \mathcal{E}(w)\leq\int_\Omega\theta(x) w(x)^p dx. \end{equation} If $\theta\neq\hat\lambda_1$, then \cite[Lemma 4.11]{MuPa} forces $w=0$. From \eqref{weakminus}--\eqref{deftheta} it follows $\| w_n\|\to 0$. However, this is impossible. So, suppose $\theta=\hat\lambda_1$. Gathering \eqref{ineqE} and $({\rm p}_2)$ together leads to $w=t\hat u_1$ for some $t\geq 0$. The above reasoning shows that $t>0$. Hence, $w\in\operatorname{int}(C_+)$. By the definition of $\{w_n\}$ we actually have $u_n^-(x)\to+\infty$ for every $x\in\Omega$. Since (A4) easily yields $$ \lim_{\xi\to-\infty}[\hat\lambda_1|\xi|^p-pF(x,\xi)] =+\infty\quad\text{uniformly in }x\in\Omega $$ (cf. Remark \ref{ffour}), Fatou's lemma gives \begin{equation}\label{abstwo} \lim_{n\to+\infty}\int_\Omega[\hat\lambda_1(u_n^-)^p-pF(x,-u_n^-(x))] dx=+\infty. \end{equation} On the other hand, via \eqref{boundminus}, besides \eqref{p1}, we get $$ \int_\Omega[\hat\lambda_1 u_n^-(x)^p-pF(x,-u_n^-(x))] dx \leq pc_1\quad\forall n\in\mathbb{N}, $$ against \eqref{abstwo}. Therefore, the sequence $\{u_n^-\}\subseteq X$ is bounded. Using \eqref{boundminus} again one sees that $\{ u_n^+\}$ enjoys the same property, which contradicts \eqref{coercivity}. Let $u_2\in X$ satisfy \[ \inf_{u\in X}\tilde\varphi(u)=\tilde\varphi(u_2). \] Arguing as in the proof of Theorem \ref{firstsol} we achieve $u_2\leq 0$ and $u_2\neq 0$. So, $u_2$ solves problem \eqref{prob} and belongs to $(-C_+)\setminus\{0\}$ by standard nonlinear regularity results. Finally, (A1) and (A4) provide $\tilde\mu>\| a\|_\infty$ such that $$ f(x,t)+\tilde\mu|t|^{p-2}t\leq 0,\quad (x,t)\in\Omega\times\mathbb{R}^-_0\,. $$ Consequently, $$ \Delta_p(-u_2)+(a+\tilde\mu)|u_2|^{p-2}u_2=f(x,u_2)+\tilde\mu|u_2|^{p-2}u_2\leq 0, $$ whence $$ \Delta_p(-u_2)\leq(a+\tilde\mu)(-u_2)^{p-1}\quad\text{in}\quad\Omega. $$ Through \cite[Theorem 5]{Va} this implies $-u_2\in\operatorname{int}(C_+)$, as desired. \end{proof} \subsection{Extremal constant-sign and nodal solutions}\label{sect3.3} The following stronger version of (A5) will be used. \begin{itemize} \item[(A5')] There exist $q\in(1,p)$, $a_4>0$, and $\delta_1>0$ such that $$ a_4|\xi|^q\leq f(x,\xi)\xi\leq qF(x,\xi)\quad\forall (x,\xi)\in\Omega\times[-\delta_1,\delta_1]. $$ \end{itemize} It plays a crucial role in getting useful information on the critical groups of $\varphi$ at zero. Precisely, the result below, whose proof is analogous to that of \cite[Proposition 4.1]{PaWi} (cf. also \cite[Theorem 3.6]{MaMoPa}), holds. \begin{lemma}\label{ckzero} Suppose \eqref{abeta}, {\rm (A1), (A5')} hold and $K(\varphi)$ is a finite set. Then $C_k(\varphi,0)=0$ for all $k\in\mathbb{N}_0$. \end{lemma} Combining (A1) with (A5') we obtain \begin{equation}\label{lowest} f(x,t)t\geq a_4|t|^q-a_5|t|^r\quad\text{in }\Omega\times\mathbb{R} \end{equation} for an appropriate $a_5>0$. Consider the auxiliary problem \begin{equation}\label{auxprob} \begin{gathered} -\Delta_p u+a(x)|u|^{p-2}u=a_4|u|^{q-2}u-a_5|u|^{r-2}u \quad \text{in }\Omega,\\ \frac{\partial u}{\partial n_p}+\beta(x) |u|^{p-2}u=0 \quad \text{on } \partial\Omega. \end{gathered} \end{equation} Note that if $u$ is a solution then $-u$ also solves this problem. \begin{lemma}\label{uplus} If \eqref{abeta} holds then \eqref{auxprob} admits a unique positive solution $u_+\in\operatorname{int}(C_+)$. \end{lemma} \begin{proof} The $C^1$-functional $\psi:X\to\mathbb{R}$ given by $$ \psi(u):=\frac{1}{p}\left(\mathcal{E}(u)+\bar a\| u^-\|_p^p\right) -\frac{a_4}{q}\| u^+\|_q^q+\frac{a_5}{r}\| u^+\|_r^r\,,\quad u\in X, $$ is coercive. Indeed, recalling that $\beta\geq 0$, $\bar a\geq\| a\|_\infty$, and $q0$ small enough. As in the proof of Theorem \ref{firstsol} we next get $u_+\geq 0$. Hence, by standard nonlinear regularity results, $u_+\in C_+\setminus\{0\}$. The conclusion $u_+\in\operatorname{int}(C_+)$ easily derives from $$ \Delta_p u_+\leq \left(\| a\|_\infty+a_5\| u_+\|_\infty^{r-p}\right) u_+^{p-1} \leq c_6u_+^{p-1}; $$ cf. \cite[Theorem 5]{Va}. Let us now come to uniqueness. Suppose $\hat u\in \operatorname{int}(C_+)$ is another solution of \eqref{auxprob}. For $u\in L^1(\Omega)$, we put \[ J(u):=\begin{cases} \frac{1}{p}\left(\|\nabla u^{1/p}\|_p^p+\int_{\partial\Omega} au\, d\sigma\right) & \text{if } u\geq 0,\; u^{1/p}\in X,\\ +\infty & \text{otherwise.}\\ \end{cases} \] \cite[Lemma 1 ]{DiSa} ensures that $J:L^1(\Omega)\to\mathbb{R}\cup\{+\infty\}$ is proper, convex, and lower semi-continuous. A simple computation, chiefly based on \cite[Theorem 2.4.54]{GaPaNA}, yields $$ J'(u_+^p)(v)=\frac{1}{p}\int_\Omega\frac{-\Delta_p u_+}{u_+^{p-1}}\, v\, dx\,,\quad J'(\hat u^p)(v)=\frac{1}{p}\int_\Omega\frac{-\Delta_p\hat u}{\hat u^{p-1}}\, v\, dx \quad\forall v\in C^1(\overline{\Omega}), $$ while the monotonicity of $J'$ leads to $$ \int_\Omega\Big( \frac{-\Delta_p u_+}{u_+^{p-1}} -\frac{-\Delta_p\hat u}{\hat u^{p-1}}\Big)\left( u_+^p-\hat u^p\right) dx\geq 0. $$ Therefore, $$ \int_\Omega\Big[a_4\Big(\frac{1}{u_+^{p-q}}-\frac{1}{\hat u^{p-q}}\Big) -a_5( u_+^{r-p}-\hat u^{r-p}) \Big] \left( u_+^p-\hat u^p\right)dx\geq 0, $$ which implies $u_+=\hat u$, because $qu_*(x), \end{cases} \\ \hat f_\pm(x,t):=\hat f(x,t^\pm), \\ \hat F(x,\xi):=\int_0^\xi \hat f(x,t)dt,\quad \hat F_{\pm}(x,\xi):=\int_0^\xi\hat f_\pm(x,t)\, dt. \end{gathered} \end{equation} It is evident that the corresponding truncated functionals \begin{equation}\label{hatphi} \begin{gathered} \hat\varphi(u):=\frac{1}{p}\left(\mathcal{E}(u)+\bar a\| u\|^p_p\right) -\int_\Omega\hat F(x,u(x))\, dx,\quad u\in X, \\ \hat\varphi_{\pm}(u):=\frac{1}{p}\left(\mathcal{E}(u)+\bar a\| u\|^p_p\right) -\int_\Omega\hat F_\pm(x,u(x))\, dx,\quad u\in X, \end{gathered} \end{equation} belong to $C^1(X)$. Moreover, by construction, one has % \begin{equation}\label{critset} K(\hat\varphi)\subseteq [v_*,u_*],\quad K(\hat\varphi_-)=\{0,v_*\},\quad K(\hat\varphi_+)=\{0,u_*\}; \end{equation} see, e.g., \cite[Lemma 3.1]{MaPaMFM}. \begin{theorem}\label{nodal} If \eqref{abeta}, {\rm (A1)--(A4), (A5'), (A6)} hold, then \eqref{prob} possesses a nodal solution $u_3\in [v_*,u_*]\cap C^1(\overline{\Omega})$. \end{theorem} \begin{proof} $X$ compactly embeds in $L^p(\Omega)$ while the Nemitskii operator $N_{\hat f_+}$ turns out to be continuous on $L^p(\Omega)$. Thus, a standard argument ensures that $\hat\varphi_+$ is weakly sequentially lower semi-continuous. Since, on account of \eqref{truncf}, it is coercive, we obtain \[ \inf_{u\in X}\hat\varphi_+(u)=\hat\varphi_+(u_0) \] for some $u_0\in X$. Reasoning as in the proof of Theorem \ref{firstsol} produces $u_0\in\operatorname{int}(C_+)$ and, by \eqref{critset}, $u_0=u_*$. Since $\hat\varphi\lfloor_{C_+}=\hat\varphi_+\lfloor_{C_+}$, the function $u_*$ turns out to be a $C^1(\overline{\Omega})$-local minimizer for $\hat\varphi$. Now, \cite[Proposition 3]{PaRa} guarantees that the same remains true with $X$ in place of $C^1(\overline{\Omega})$. A similar argument applies to $v_*$. Consequently, $u_*,v_*$ are local minimizer for $\hat\varphi$. We may assume $K(\hat\varphi)$ finite, otherwise infinitely many nodal solutions do exist by \eqref{critset}. Let $\hat\varphi(v_*)\leq\hat\varphi(u_*)$ (the other case is analogous). Without loss of generality, the local minimizer $u_*$ for $\hat\varphi$ can be supposed proper. Thus, there exists $\rho\in(0,\| u_*-v_*\|)$ such that \begin{equation}\label{crho} \hat\varphi(u_*)\hat\lambda_2$ and $a_7>0$ such that $$ a_6\leq\liminf_{t\to 0}\frac{f(x,t)}{|t|^{p-2}t} \leq\limsup_{t\to 0}\frac{f(x,t)}{|t|^{p-2}t}\leq a_7 $$ uniformly in $x\in\Omega$. \end{itemize} Via (A1) and (A5'') one has \[ f(x,t)t\geq a_8|t|^p-a_9|t|^r,\quad (x,t)\in\Omega\times\mathbb{R}, \] for appropriate $a_8>\hat\lambda_2$, $a_9>0$. Consider the auxiliary problem \begin{equation}\label{auxprobone} \begin{gathered} -\Delta_p u+a(x)|u|^{p-2}u=a_8|u|^{p-2}u-a_9|u|^{r-2}u \quad \text{in }\Omega,\\ \frac{\partial u}{\partial n_p}+\beta(x) |u|^{p-2}u=0 \quad \text{on } \partial\Omega. \end{gathered} \end{equation} Note that if $u$ is a solution then $-u$ also solves this problem. Reasoning as above we see that: \begin{itemize} \item Problem \eqref{auxprobone} admits a unique positive solution $u_+\in\operatorname{int}(C_+)$. \item $u_-:=-u_+$ represents the unique negative solution of \eqref{auxprobone}. \item Under assumptions {\rm (A1)--(A4), (A5''), (A6)} and \eqref{abeta}, problem \eqref{prob} possesses both a smallest positive solution $u_*$ and a biggest negative solution $v_*$. Further, $-v_*,u_*\in\operatorname{int}(C_+)$. \end{itemize} Now, the same arguments used in the proof of \cite[Theorem 3.3]{MaPaMFM} yield the following result. \begin{theorem} Let \eqref{abeta}, {\rm (A1)--(A4), (A5'')}, and {\rm (A6)} be satisfied. Then \eqref{prob} admits a nodal solution $u_3\in [v_*,u_*]\cap C^1(\overline{\Omega})$. \end{theorem} \subsection{Existence of at least four nontrivial solutions} Gathering the results in Sections 3.1--3.3 we directly obtain the next one. \begin{theorem}\label{firstmultiple} If \eqref{abeta}, {\rm (A1)--(A4), (A5')--(A6)} hold, then \eqref{prob} possesses at least four solutions $u_0,u_1\in\operatorname{int}(C_+)$, $u_2\in-\operatorname{int}(C_+)$, and $u_3\in [u_2,u_0]\cap C^1(\overline{\Omega})$ nodal. Moreover, $u_0\leq u_1$. \end{theorem} \begin{remark}\label{othercase} \rm Hypothesis (A5') can be substituted by (A5'') without changing the conclusion. \end{remark} \section{Semilinear case} From now on we shall assume $p=2$. Then the regularity results of \cite{Wa} allow to weaken \eqref{abeta} as follow, see \cite{DMP, MaPaJMAA2016}, \begin{equation}\label{abetaone} a\in L^s(\Omega)\text{ for some $s>N$, } a^+\in L^\infty(\Omega),\quad\beta\in W^{1,\infty}(\partial\Omega),\text{ and } \beta\geq 0. \end{equation} Further, the energy functional $\varphi$ given by \eqref{defphi} fulfills condition (C) once \eqref{abetaone}, (A1), (A2), and (A4) hold; see Proposition \ref{cerami}. \begin{lemma}\label{ckzerosem} Under assumptions \eqref{abetaone}, {\rm (A1)}, and \begin{itemize} \item[(A7)] $\hat\lambda_m t^2\leq f(x,t)t\leq\hat\lambda_{m+1}t^2$ in $\Omega\times [-\delta_2,\delta_2]$, with appropriate $m\in\mathbb{N}$, $\delta_2>0$, \end{itemize} one has $$ C_k(\varphi,0)=\delta_{k,d_m}\mathbb{Z}\quad\forall k\in\mathbb{N}_0\,, $$ where $d_m:=\dim (\bar H_m)$, provided $\varphi$ satisfies $(C)$ and $0\in K(\varphi)$ is isolated. \end{lemma} \begin{proof} It is similar to that of \cite[Lemma 3.3]{DMP}. So, we only sketch the main points. Pick a $\theta\in (\hat\lambda_m,\hat\lambda_{m+1})$ and define $$ \psi(u):=\frac{1}{2}\left(\mathcal{E}(u)-\theta\| u\|_2^2\right),\quad u\in X\,. $$ Thanks to (A7), zero is a non-degenerate critical point of $\psi$ having Morse index $d_m$, which entails $$ C_k(\psi,0)=\delta_{k,d_m}\mathbb{Z}\quad\forall k\in\mathbb{N}_0\,; $$ see \eqref{kd}. Now, recall that every $v\in X$ admits a unique sum decomposition $v=\bar v+\hat v$, with $\bar v\in \bar H_m$, $\hat v\in \overline{\hat H_{m+1}}$. If $u\in C^1(\overline{\Omega})$ and $0<\| u\|_{ C^1(\overline{\Omega})}<\delta_2$ then \begin{equation}\label {hatbarone} \langle\varphi'(u),\hat u-\bar u\rangle=\mathcal{E}(\hat u)-\mathcal{E}(\bar u) -\int_\Omega f(x,u)(\hat u-\bar u)\, dx\,. \end{equation} By (A7) again, one arrives at $$ f(x,u)(\hat u-\bar u)=\frac{f(x,u)}{u}u(\hat u-\bar u) \leq \begin{cases} \hat\lambda_{m+1}(\hat u^2-\bar u^2) & \text{if } u(\hat u-\bar u)\geq 0,\\ -\hat\lambda_m(\bar u^2-\hat u^2) & \text{otherwise}. \end{cases} $$ Hence, \begin{equation}\label{hatbartwo} f(x,u(x))(\hat u(x)-\bar u(x))\leq\hat\lambda_{m+1}\hat u(x)^2 -\hat\lambda_m\bar u(x)^2\quad\text{in }\Omega\,. \end{equation} From \eqref{hatbarone}, \eqref{hatbartwo}, and \eqref{p3} it follows that \[ \langle\varphi'(u),\hat u-\bar u\rangle\geq \mathcal{E}(\hat u)-\hat\lambda_{m+1}\|\hat u\|_2^2-[\mathcal{E}(\bar u)-\hat\lambda_m\|\bar u\|_2^2]\geq 0\,. \] Using \cite[Lemma 2.2]{DMP} we obtain \[ \langle\psi'(u),\hat u-\bar u\rangle =\mathcal{E}(\hat u)-\theta\|\hat u\|_2^2 -[\mathcal{E}(\bar u)-\theta\|\bar u\|_2^2] \geq c_1\| u\|^2 \] for some $c_1>0$. Therefore, the homotopy $$ h(t,v):=(1-t)\varphi(v)+t\psi(v),\quad (t,v)\in [0,1]\times X $$ fulfills the inequality $$ \langle h'_v(t,u),\hat u-\bar u\rangle\geq tc_1\| u\|^2 \quad\forall t\in [0,1]\,, $$ and \cite[Theorem 5.2]{CH} can be applied. By that result $C_k(\varphi,0)=C_k(\psi,0)$, which completes the proof. \end{proof} The same arguments made in \cite[Proposition 15]{PaRa1} yield the next result. \begin{lemma}\label{ckinfty} Assume \eqref{abetaone}, {\rm (A1)}, and {\rm (A2)} hold. If $\varphi$ satisfies {\rm (C)} and is bounded below on $K(\varphi)$, then $C_k(\varphi,\infty)=0$ for all $k\in\mathbb{N}_0$. \end{lemma} The condition below will take the place of (A1). \begin{itemize} \item[(A1')] $f(x,\cdot)\in C^1(\mathbb{R})$ for every $x\in\Omega$. There exist $a_1\in L^\infty(\Omega)$, $r\in (2,2^*)$ such that $$ |f'_t(x,t)|\leq a_1(x)(1+|t|^{r-2})\quad\forall (x,t)\in\Omega\times\mathbb{R}. $$ \end{itemize} \begin{remark}\label{stronger} \rm An easy computation shows that (A1') implies (A6). \end{remark} We are now in a position to establish a five-solutions existence result. It complements those previously obtained in \cite{DMP, MaPaJMAA2016}. \begin{theorem}\label{fivesol} Let \eqref{abetaone}, {\rm (A1'), (A2)--(A4)} be satisfied. Suppose also that \begin{itemize} \item[ (A7')] either $$ a_{10}t^2\leq f(x,t)t\leq\hat\lambda_3t^2,\quad (x,t)\in\Omega\times [-\delta_3,\delta_3], $$ for some $a_{10}>\hat\lambda_2$ and $\delta_3>0$, or $$ \hat\lambda_m t^2\leq f(x,t)t\leq\hat\lambda_{m+1}t^2,\quad (x,t)\in\Omega\times [-\delta_3,\delta_3], $$ where $m\geq 3$. \end{itemize} Then \eqref{probp=2} possesses at least five nontrivial solutions $u_i\in C^1(\overline{\Omega})$, $i=0,\ldots,4$, with $u_0$, $u_1$, $u_2$, $u_3$ as in Theorem \ref{firstmultiple}. \end{theorem} \begin{proof} Thanks to Remarks \ref{othercase} and \ref{stronger}, the conclusion of Theorem \ref{firstmultiple} holds for the present framework. So, it remains to find a further solution $u_4\in C^1(\overline{\Omega})\setminus\{0\}$. Without loss of generality, we assume that $u_0$, $u_3$ are extremal (see Section \ref{sect3.3}), while a standard argument based on (A6) and \eqref{abetaone} yields $u_3\in\operatorname{int}_{C^1(\overline{\Omega})}([u_2,u_0])$; vide, e.g., \cite[Theorem 3.2]{MaPaJMAA2016}. Still we write $\hat f$ for the function defined in \eqref{truncf} but with $v_*$ and $u_*$ replaced by $u_2$ and $u_0$, respectively. \cite[Lemma 2.1]{DMP} provides $\hat a,\hat b>0$ fulfilling $$ \mathcal{E}(u)+\hat a\| u\|_2^2\geq\hat b\| u\|^2\quad\forall u\in X\,. $$ Pick any $\bar a\geq\hat a$ and consider the functional $\hat\varphi$ given by \eqref{hatphi}. The same reasoning adopted in the proof of Theorem \ref{nodal} ensures here that $C_k(\hat\varphi, u_3)=C_k(\varphi,u_3)$. Thus $$ C_1(\varphi,u_3)\neq 0\,, $$ because $u_3$ is a mountain-pass type critical point for $\hat\varphi$; cf. \cite[Corollary 6.81]{MoMoPa}. By (A1') one has $\varphi\in C^2(X)$ as well as \begin{equation}\label{secondderivative} \langle\varphi''(u_3)u,v\rangle=\int_\Omega(\nabla u\cdot\nabla v+auv)dx +\int_{\partial\Omega}\beta uv\, d\sigma- \int_\Omega f'_t(x,u_3)uvdx, \end{equation} for $ u,v\in X$. Hence, if the Morse index of $u_3$ is zero, then \begin{equation}\label{secondcons} \|\nabla u\|_2^2+\int_{\partial\Omega}\beta u^2d\sigma \geq\int_\Omega [f'_t(x,u_3)-a]u^2dx\quad\forall u\in X. \end{equation} Write $\alpha:=[f'_t(x,u_3)-a]^+$ and observe that $\alpha\in L^s(\Omega)$. Two situations may occur. \smallskip \noindent(1) $\alpha=0$. Due to \eqref{secondderivative}, for every $u\in {\rm ker}\varphi''(u_3)$ we get $$ \|\nabla u\|_2^2+\int_{\partial\Omega}\beta(x) u(x)^2d\sigma\leq 0, $$ which implies $u$ constant. \smallskip \noindent(2) $\alpha\neq 0$. From \eqref{secondcons} it follows $\hat\lambda_1(\alpha)\geq 1$ and by \eqref{secondderivative} the assertion ${\rm ker}\varphi''(u_3)\neq\{0\}$ forces $\hat\lambda_1(\alpha)=1$, whence $\dim \,{\rm ker}\varphi''(u_3)=1$. In both cases we arrive at $\dim \,{\rm ker}\varphi''(u_3)\leq 1$. So, on account of \cite[Proposition 6.101]{MoMoPa}, \begin{equation}\label{ckuthree} C_k(\varphi, u_3)=\delta_{k,1}\mathbb{Z}\quad\forall k\in\mathbb{N}_0. \end{equation} Next, we define $$ \varphi_+(u):=\frac{1}{2}\mathcal{E}(u)-\int_\Omega F_+(x,u(x))\, dx,\quad u\in X, $$ where $F_+(x,\xi):=\int_0^\xi f(x,t)^+\, dt$. Assumption (A7) easily leads to $\varphi\lfloor_{C_+}=\varphi_+\lfloor_{C_+}$, which entails $$ C_k(\varphi\lfloor_{C^1(\overline{\Omega})},u_1) =C_k(\varphi_+\lfloor_{C^1(\overline{\Omega})},u_1) $$ because $u_1\in \operatorname{int}(C_+)$; see Theorem \ref{secondsol}. By denseness one has $C_k(\varphi,u_1)=C_k(\varphi_+, u_1)$. Now, observe that $\varphi_+=\varphi_0+c$, with appropriate $c>0$ and $\varphi_0$ as in \eqref{phizero}, on a neighbourhood of $u_1$. Consequently, $C_k(\varphi_+, u_1)=C_k(\varphi_0,u_1)$. Since $u_1$ is a mountain-pass type critical point for $\varphi_0$ (cf. the proof of Theorem \ref{secondsol}), the same argument made above gives \begin{equation}\label{ckuone} C_k(\varphi, u_1)=\delta_{k,1}\mathbb{Z}\,,\quad k\in\mathbb{N}_0. \end{equation} Gathering Theorem \ref{secondsol} and \cite[Proposition 6.95]{MoMoPa}, we derive \begin{equation}\label{ckuzero} C_k(\varphi, u_0)=\delta_{k,0}\mathbb{Z}\quad\forall k\in\mathbb{N}_0. \end{equation} Likewise, \begin{equation}\label{ckutwo} C_k(\varphi, u_2)=\delta_{k,0}\mathbb{Z}\,,\quad \forall k\in\mathbb{N}_0, \end{equation} while Lemmas \ref{ckzerosem}--\ref{ckinfty} yield \begin{equation}\label{ckfinal} C_k(\varphi,0)=\delta_{k,d_m}\mathbb{Z}\,,\quad C_k(\varphi,\infty)=0\quad\forall k\in\mathbb{N}_0. \end{equation} Finally, if $K(\varphi)=\{0, u_0,u_1,u_2,u_3\}$ then \eqref{morse}, with $t=-1$, and \eqref{ckuthree}--\eqref{ckfinal} would imply $$ (-1)^{d_m}+2(-1)^0+2(-1)^1=0, $$ which is impossible. Thus, there exists $u_4\in K(\varphi)\setminus\{0, u_0,u_1,u_2,u_3\}$, i.e., a fifth nontrivial solution to \eqref{prob}. Standard regularity results \cite{Wa} ensure that $u_4\in C^1(\overline{\Omega})$. \end{proof} \subsection*{Acknowledgements} This research was done under the auspices of GNAMPA of INDAM and within the 2016--2018 Research Plan - Intervention Line 2: `Variational methods and differential equations'. \begin{thebibliography}{00} \bibitem{APS2} S. Aizicovici, N.~S. Papageorgiou, V. Staicu; \emph{Degree Theory for Operators of Monotone Type and Nonlinear Elliptic Equations with Inequality Constraints}, Mem. Amer. Math. Soc., \textbf{196} (2008). \bibitem{APS1} S. Aizicovici, N.~S. Papageorgiou, V. Staicu; \emph{Existence of multiple solutions with precise sign information for superlinear Neumann problems,} Ann. Mat. Pura Appl., \textbf{188} (2009), 679--710. \bibitem{ArRu} D. Arcoya and D. Ruiz; \emph{The Ambrosetti-Prodi problem for the p-Laplace operator,} Comm. 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