\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2018 (2018), No. 41, pp. 1--10.\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-2018/41\hfil Ambrosetti-Prodi problem with degenerate potential] {Ambrosetti-Prodi problem with degenerate potential and Neumann boundary condition} \author[D. D. Repov\v{s} \hfil EJDE-2018/41\hfilneg] {Du\v{s}an D. Repov\v{s}} \address{Du\v{s}an D. Repov\v{s} \newline Faculty of Education and Faculty of Mathematics and Physics, University of Ljubljana, SI-1000 Ljubljana, Slovenia} \email{dusan.repovs@guest.arnes.si} \dedicatory{Communicated by Vicentiu D. Radulescu} \thanks{Submitted July 20, 2017. Published February 6, 2018.} \subjclass[2010]{35J65, 35J25, 58E07} \keywords{Ambrosetti-Prodi problem; degenerate potential; topological degree; \hfill\break\indent anisotropic continuous media} \begin{abstract} We study the degenerate elliptic equation $$ -\operatorname{div}(|x|^\alpha\nabla u) =f(u)+t\phi(x)+h(x) $$ in a bounded open set $\Omega$ with homogeneous Neumann boundary condition, where $\alpha\in(0,2)$ and $f$ has a linear growth. The main result establishes the existence of real numbers $t_*$ and $t^*$ such that the problem has at least two solutions if $t\leq t_*$, there is at least one solution if $t_*t^*$. The proof combines \emph{a priori} estimates with topological degree arguments. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \allowdisplaybreaks \section{Introduction} Let $\Omega\subset\mathbb{R}^N$ be a bounded open set with smooth boundary. In their seminal paper \cite{ap}, Ambrosetti and Prodi studied the semilinear elliptic problem \begin{equation} \label{pr1} \begin{gathered} \Delta u+f(u)=v(x) \quad\text{in } \Omega\\ u=0 \quad\text{on } \partial\Omega, \end{gathered} \end{equation} where the nonlinearity $f$ is a function whose derivative crosses the first (principal) eigenvalue $\lambda_1$ of the Laplace operator in $H^1_0(\Omega)$, in the sense that $$ 0<\lim_{t\to -\infty}\frac{f(t)}{t}<\lambda_1 <\lim_{t\to +\infty}\frac{f(t)}{t}<\lambda_2. $$ By using the abstract approach developed in \cite{ap}, Ambrosetti and Prodi have been able to describe the exact number of solutions of \eqref{pr1} in terms of $v$, provided that $f''>0$ in $\mathbb{R}$. More precisely, they proved that there exists a closed connected manifold $A_1\subset C^{0,\alpha}(\overline\Omega)$ of codimension 1 such that $C^{0,\alpha}(\overline\Omega)\setminus A_1=A_0\cup A_2$ and problem \eqref{pr1} has exactly zero, one or two solutions according as $v$ is in $A_0$, $A_1$, or $A_2$. The proof of this pioneering result is based upon an extension of Cacciopoli's mapping theorem to some singular case. A cartesian representation of $A_1$ is due to Berger and Podolak \cite{berger}, who observed that it is convenient to write problem \eqref{pr1} in an equivalent way, as follows. Let $$ Lu:=\Delta u+\lambda_1u,\qquad g(u):=f(u)-\lambda_1u $$ and $$ v(x):=t\phi(x)+h(x)\quad\text{with}\quad \int_\Omega h(x)\phi(x)\,dx=0. $$ In such a way, problem \eqref{pr2} is equivalent to \begin{equation}\label{pr2} \begin{gathered} Lu+g(u)=t\phi(x)+h(x) \quad\text{in } \Omega\\ u=0\quad\text{on } \partial\Omega, \end{gathered} \end{equation} with $g''>0$ in $\mathbb{R}$ and $$ -\lambda_1<\lim_{t\to -\infty}\frac{g(t)}{t} <0<\lim_{t\to +\infty}\frac{g(t)}{t}<\lambda_2-\lambda_1. $$ Under these assumptions, Berger and Podolak \cite{berger} proved that there exists $t_1$ such that problem \eqref{pr2} has exactly zero, one or two solutions according as $tt_1$. The proof of this result is based on a global Lyapunov-Schmidt reduction method. For related developments on Ambrosetti-Prodi problems we refer to Amann and Hess \cite{amann}, Arcoya and Ruiz \cite{arcoya}, Dancer \cite{dancer}, Hess \cite{hess}, Kazdan and Warner \cite{kazdan}, Mawhin \cite{maw1, maw2}. The present paper is concerned with the Ambrosetti-Prodi problem in relationship with the contributions of Caldiroli and Musina \cite{caldi}, who initiated the study of Dirichlet elliptic problems driven by the differential operator $\operatorname{div} (|x|^\alpha\nabla u)$, where $\alpha\in (0,2)$. This operator is a model for equations of the type \begin{equation}\label{bao} -\operatorname{div} (a(x)\nabla u)=f(x,u)\quad x\in\Omega, \end{equation} where the weight $a$ is a non-negative measurable function that is allowed to have ``essential" zeros at some points or even to be unbounded. According to Dautray and Lions \cite[p. 79]{dautray}, equations like \eqref{bao} are introduced as models for several physical phenomena related to equilibrium of anisotropic continuous media which possibly are somewhere ``perfect" insulators or ``perfect" conductors. We also refer to the works by Murthy and Stampacchia \cite{franchi}, by Baouendi and Goulaouic \cite{bg} concerning degenerate elliptic operators (regularity of solutions and spectral theory). Problem \eqref{bao} also has some interest in the framework of optimization and $G$-convergence, cf. Franchi, Serapioni, and Serra Cassano \cite{franchi}. For degenerate phenomena in nonlinear PDEs we also refer to Fragnelli and Mugnai \cite{frag}, and Nursultanov and Rozenblum \cite{nurs}. This article studies of the Ambrosetti-Prodi problem in the framework of the degenerate elliptic operator studied in \cite{caldi}. A feature of this work is that the analysis is developed in the framework of Neumann boundary conditions. \section{Main result and abstract setting} Let $\alpha\in (0,2)$ and let $\Omega\subset\mathbb{R}^N$ be a bounded open set with smooth boundary. Consider the nonlinear problem \begin{equation}\label{1} \begin{gathered} -\operatorname{div}(|x|^\alpha\nabla u) =f(u)+t\phi(x)+h(x) \quad\text{in } \Omega\\ \frac{\partial u}{\partial\nu}=0\quad\text{on } \partial\Omega. \end{gathered} \end{equation} We assume that $f:\mathbb{R}\to\mathbb{R}$ is a continuous function such that \begin{equation}\label{f} \limsup_{t\to -\infty}\frac{f(t)}{t} <0<\liminf_{t\to +\infty}\frac{f(t)}{t} \end{equation} and there exists $C_f>0$ such that \begin{equation}\label{f1} |f(t)|\leq C_f(1+|t|) \quad \text{for all $t\in\mathbb{R}$}. \end{equation} Since the first eigenvalue of the Laplace operator with respect to the Neumann boundary condition is zero, condition \eqref{f} asserts that the nonlinear term $f$ crosses this eigenvalue. Next, we assume that $\phi$, $h\in L^\infty(\Omega)$ and \begin{equation}\label{phi} \phi\geq 0,\quad \phi\not\equiv 0\quad \text{in } \Omega. \end{equation} Since $\alpha>0$, the weight $|x|^\alpha$ breaks the invariance under translations and can give rise to an abundance of existence results, according to the geometry of the open set $\Omega$. For $\zeta\in C^\infty_c(\Omega)$ we define $$ \|\zeta\|^2_\alpha:=\int_\Omega (|x|^\alpha|\nabla\zeta|^2+\zeta^2)\,dx $$ and we consider the function space $$ H^1(\Omega;|x|^\alpha):=\text{closure of $C^\infty_c(\overline{\Omega})$ with respect to the $\|\cdot\|_\alpha$-norm}. $$ It follows that $ H^1(\Omega;|x|^\alpha)$ is a Hilbert space with respect to the scalar product $$ \langle u,v\rangle_\alpha:=\int_\Omega (|x|^\alpha\nabla u\cdot\nabla v+uv)\,dx, \quad\text{for all $u.v\in H^1(\Omega;|x|^\alpha)$}. $$ Moreover, by the Caffarelli-Kohn-Nirenberg inequality (see \cite[Lemma 1.2]{caldi}), the space $ H^1(\Omega;|x|^\alpha)$ is continuously embedded in $L^{2_\alpha^*}(\Omega)$, where $2_\alpha^*$ denotes the corresponding critical Sobolev exponent, that is, $2_\alpha^*=2N/(N-2+\alpha)$. We say that $u$ is a solution of problem \eqref{1} if $u\in H^1(\Omega;|x|^\alpha)$ and for all $v\in H^1(\Omega;|x|^\alpha)$ $$ \int_\Omega |x|^\alpha\nabla u\cdot\nabla v\,dx =\int_\Omega f(u)v\,dx+t\int_\Omega \phi v\,dx+\int_\Omega hv\,dx. $$ Since the operator $Lu:=-\operatorname{div}(|x|^\alpha\nabla u)$ is uniformly elliptic on any strict subdomain $\omega$ of $\Omega$ with $0\notin\overline\omega$, the standard regularity theory can be applied in $\omega$. Hence, a solution $u\in H^1(\Omega;|x|^\alpha)$ of problem \eqref{1} is of class $C^\infty$ on $\Omega\setminus\{0\}$. We refer to Brezis \cite[Theorem IX.26]{brezis} for more details. The main result of this paper extends to the degenerate setting formulated in problem \eqref{1} the abstract approach developed by Hess \cite{hess} and de Paiva and Montenegro \cite{depaiva}. For related properties on Ambrosetti-Prodi problems with Neumann boundary condition, we refer to Presoto and de Paiva \cite{presoto}, Sovrano \cite{sovrano}, V\'elez-Santiago \cite{velez, velez1}. \begin{theorem}\label{th1} Assume that hypotheses \eqref{f}, \eqref{f1} and \eqref{phi} are fulfilled. Then there exist real numbers $t_*$ and $t^*$ with $t_*\leq t^*$ such that the following properties hold: \begin{itemize} \item[(a)] problem \eqref{1} has at least two solutions solution, provided that $t\leq t_*$; \item[(b)] problem \eqref{1} has at least one solution, provided that $t_*t^*$. \end{itemize} \end{theorem} \subsection*{Strategy of the proof} Let $C_f$ be the positive constant defined in hypothesis \eqref{f1} and assume that $v\in L^2(\Omega)$. Consider the linear Neumann problem \begin{equation}\label{2} \begin{gathered} -\operatorname{div}(|x|^\alpha\nabla w)+C_fw =v \quad\text{in } \Omega\\ \frac{\partial w}{\partial\nu}=0\quad\text{on } \partial\Omega. \end{gathered} \end{equation} With the same arguments as in \cite[Chapter IX, Exemple 4]{brezis}, problem \eqref{2} has a unique solution $w\in H^1(\Omega;|x|^\alpha)$. This defines a linear map $$ L^2(\Omega)\ni v\mapsto w\in H^1(\Omega;|x|^\alpha). $$ It follows that the linear operator $T:L^\infty(\Omega)\to H^1(\Omega;|x|^\alpha)$ defined by $Tv:=w$ is compact. We also point out that if $v\geq 0$ then $w\geq 0$, hence $T$ is a positive operator. We observe that $u$ is a solution of problem \eqref{1} if and only if $u$ is a fixed point of the nonlinear operator $$ S_t(v):=T(f(v)+C_fv+t\phi +h). $$ Thus, solving problem \eqref{1} reduces to finding the critical points of $S_t$. \section{Proof of the main result} We split the proof into several steps. \subsection{Non-existence of solutions if $t$ is big}\label{s1} In fact, we show that a necessary condition for the existence of solutions of problem \eqref{1} is that the parameter $t$ should be small enough. We first observe that hypothesis \eqref{f} implies that there are positive constants $C_1$ and $C_2$ such that $$ f(t)\geq C_1|t|-C_2\quad\text{for all $t\in\mathbb{R}$}. $$ Assuming that $u$ is a solution of problem \eqref{1}, we obtain by integration \begin{align*} 0& =\int_\Omega f(u)\,dx+t\int_\Omega\phi \,dx+\int_\Omega h\,dx\\ &\geq C_1\int_\Omega |u|\,dx-C_2|\Omega|+t\int_\Omega\phi \,dx+\int_\Omega h\,dx\\ &\geq -C_2|\Omega|+t\int_\Omega\phi \,dx+\int_\Omega h\,dx. \end{align*} It follows that a necessary condition for the existence of solutions of problem \eqref{1} is $$ t\leq\frac{ C_2|\Omega|-\int_\Omega h\,dx}{\int_\Omega\phi \,dx}\,. $$ \subsection{Problem \eqref{1} has solutions for small $t$: a preliminary step} \label{s2} In this subsection, we prove that for any $\rho>0$ there exists $t_\rho\in\mathbb{R}$ such that for all $t\leq t_\rho$ and all $s\in [0,1]$ we have \begin{equation}\label{3} v\neq sS_t(v)\quad\text{for all $v\in L^\infty(\Omega)$, $\|v^+\|_\infty=\rho$}. \end{equation} Our argument is by contradiction. Thus, there exist three sequences $(s_n)\subset [0,1]$, $(t_n)\subset\mathbb{R}$ and $(v_n)\subset L^\infty(\Omega)$ such that $\lim_{n\to\infty}t_n=-\infty$, $\|v_n^+\|_\infty=\rho$ and \begin{equation}\label{4} v_n=s_nS_{t_n}(v_n)\quad\text{for all } n\geq 1. \end{equation} By hypothesis \eqref{f1} we have \begin{equation}\label{5} \begin{aligned} f(v_n)+C_fv_n &\leq C_f+C_f|v_n|+C_fv_n\\ &= C_f+2C_fv_n^+\leq C_f+2C_f\rho. \end{aligned} \end{equation} Using the definition of $S$ and the fact that $T$ is a positive operator, relations \eqref{4} and \eqref{5} yield \begin{align*} v_n &=s_nS_{t_n}(v_n)=s_nT(f(v_n)+C_fv_n+t_n\phi +h)\\ &\leq s_nT(C_f+2C_f\rho+t_n\phi +h), \end{align*} hence $$ v_n^+\leq s_n[T(C_f+2C_f\rho+t_n\phi +h)]^+. $$ Let $$ w_n:=C_f+2C_f\rho+t_n\phi +h. $$ It follows that $w_n$ is the unique solution of the problem \begin{gather*} -\operatorname{div}(|x|^\alpha w_n)+C_fw_n =C_f+2C_f\rho+t_n\phi +h \quad\text{in } \Omega\\ \frac{\partial w_n}{\partial\nu}=0 \quad\text{on } \partial\Omega. \end{gather*} Dividing by $t_n$ (recall that $\lim_{n\to\infty}t_n=-\infty$) we obtain \begin{gather*} -\operatorname{div}\Big(|x|^\alpha \frac{w_n}{t_n}\Big)+C_f\,\frac{w_n}{t_n} =\phi +\frac{C_f+2C_f\rho +h}{t_n} \quad\text{in } \Omega\\ \frac{\partial}{\partial\nu}\Big(\frac{w_n}{t_n} \Big)=0 \quad\text{on } \partial\Omega. \end{gather*} However, $$ \lim_{n\to\infty}\frac{C_f+2C_f\rho +h}{t_n}=0. $$ So, by elliptic regularity (see \cite[Theorem IX.26]{brezis}), $$ \frac{w_n}{t_n}\to T\phi\quad\text{in } C^{1,\beta}(\overline\Omega\setminus\{0\}) \text{ as } n\to\infty. $$ Next, by the strong maximum principle, we have $T\phi >0$ in $\Omega$ and $$ \frac{\partial T\phi}{\partial\nu}(x)<0\quad\text{for all } x\in\partial\Omega \text{ with } T\phi (x)=0. $$ We deduce that for all $n$ sufficiently large $$ \frac{w_n}{t_n}>0\quad\text{in } \Omega, $$ which forces $w_n^+=0$ for all $n$ large enough. But $$ v_n^+\leq s_nw_n^+\leq w_n^+, $$ hence $$ \rho=\|v_n^+\|_\infty\leq\|w_n^+\|_\infty =0, $$ a contradiction. This shows that our claim \eqref{3} is true. \subsection{Problem \eqref{1} has solutions for small $t$: an intermediary step} \label{s3} In this subsection, we prove that for any $t\in\mathbb{R}$ there exists $\rho_t>0$ such that for all $s\in [0,1]$ we have \begin{equation}\label{33} v\neq sS_t(v)\quad\text{for all $v\in L^\infty(\Omega)$, $\|v^-\|_\infty=\rho_t$}. \end{equation} Fix arbitrarily $t\in\mathbb{R}$. Assume that there exist $s\in [0,1]$ and a function $v$ (depending on $s$) such that $v=sS_t(v)$. It follows that $v$ is the unique solution of the problem \begin{equation} \label{34} \begin{gathered} -\operatorname{div}(|x|^\alpha\nabla v)+C_fv =s(f(v)+C_fv+t\phi +h) \quad\text{in } \Omega\\ \frac{\partial v}{\partial\nu}=0 \quad\text{on } \partial\Omega. \end{gathered} \end{equation} By hypotheses \eqref{f} and \eqref{f1}, there exist positive constants $C_3$ and $C_4$ with $C_30$ such that \begin{equation}\label{38} \|w\|_\infty\leq C_0\quad\text{for all } s\in[0,1]. \end{equation} Next, relation \eqref{37} yields $$ v^-=\max\{-v,0\}\leq\max\{-w,0\}=w^-\quad\text{in}\ \Omega. $$ Using now the uniform bound established in \eqref{38}, we conclude that our claim \eqref{33} follows if we choose $\rho_t=C_0+1$. \subsection{Problem \eqref{1} has a solution for small $t$}\label{s4} Let $\rho>0$ and let $t_\rho$ be as defined in subsection \ref{s2} such that relation \eqref{3} holds. We prove that problem \eqref{1} has at least one solution, provided that $t\leq t_\rho$. Fix $t\leq t_\rho$ and let $\rho_t$ be the positive number defined in subsection \ref{s3}. Consider the open set $$ G=G_t:=\{v\in L^\infty(\Omega): \|v^+\|_\infty<\rho,\; \|v^-\|_\infty<\rho_t\}. $$ It follows that $$ v\neq sS_t(v)\quad\text{for all $v\in\partial G$, all $s\in[0,1]$}. $$ So, we can apply the homotopy invariance property of the topological degree, see Denkowski, Mig\'orski and Papageorgiou \cite[Theorem 2.2.12]{denko}. It follows that $$ \deg (I-S_t,G,0)=\deg (I,G,0)=1. $$ We conclude that $S_t$ has at least one fixed point for all $t\leq t_\rho$, hence problem \eqref{1} has at least one solution. \subsection{Proof of Theorem \ref{th1} concluded}\label{s5} We first show that problem \eqref{1} has a subsolution for all $t$. Fix a positive real number $t$. By \eqref{ctrei}, we have $$ f(u)+t\phi +h\geq -C_3u-C_4-|t|\,\|\phi\|_\infty -\|h\|_\infty\quad\text{for all}\ u\in\mathbb{R}. $$ It follows that the function $$ u\equiv -\frac{|t|\,\|\phi\|_\infty+\|h\|_\infty+C_4}{C_3} $$ is a subsolution of problem \eqref{1}. Next, with the same arguments as in the proof of \cite[Lemma 2.1]{depaiva}, we obtain that if $t$ belongs to a bounded interval $I$ then the set of corresponding solutions of problem \eqref{1} is uniformly bounded in $L^\infty(\Omega)$. Thus, there exists $C=C(I)>0$ such that for every solution of \eqref{1} corresponding to some $t\in I$ we have $\|u\|_\infty\leq C$. Since weak solutions of problem \eqref{1} are bounded, the nonlinear regularity theory of G.~Lieberman \cite{lie} implies that for every $\omega\subset\subset\Omega$ with $0\notin\overline\omega$, the set of all solutions corresponding to $I$ is bounded in $C^{1,\beta}(\overline\omega)$. We already know (subsection \ref{s1}) that problem \eqref{1} does not have any solution for large values of $t$ and solutions exist if $t$ is small enough (section \ref{s4}). Let $$ {\mathcal S}:=\{t\in\mathbb{R}: \text{problem \eqref{1} has a solution}\}. $$ It follows that ${\mathcal S}\neq \emptyset$. Let $$ t^*:=\sup{\mathcal S}<+\infty. $$ We prove in what follows that problem \eqref{1} has a solution if $t=t^*$. Indeed, by the definition of $t^*$, there is an increasing sequence $(t_n)\subset{\mathcal S}$ that converges to $t^*$. Let $u_n$ be a solution of \eqref{1} corresponding to $t=t_n$. Since $(t_n)$ is a bounded sequence, we deduce that the sequence $(u_n)$ is bounded in $C^{1,\beta}(\overline\omega)$ for all $\omega\subset\subset\Omega$ with $0\notin\overline\omega$. By the Arzela-Ascoli theorem, the sequence $(u_n)$ is convergent to some $u_*$ in $C^{1}(\overline\omega)$, which is a solution of problem \eqref{1} for $t=t^*$. Fix arbitrarily $t_00$ there exists a real number $t_\rho$ such that problem \eqref{1} has at least one solution, provided that $t\leq t_\rho$. Let $$ t_*:=\sup\{t_\rho: \rho>0\}. $$ We already know that \eqref{1} has at least one solution for all $t0$. Recall that if $t$ belongs to a bounded interval $I$ then the set of corresponding solutions of problem \eqref{1} is uniformly bounded in $L^\infty(\Omega)$. So, choosing eventually a bigger $R$, we can assume that $\|u\|_\infty0\quad\text{for every}\ z\in\overline\Omega. $$ Under this assumption, the weight $a$ could be nonsmooth, as the Taylor expansion formula can easily show. For example, the function $a$ cannot be of class $C^2$ and it cannot have bounded derivatives if $\alpha\in (0, 1)$. As established in \cite[Lemma 2.2, Remark 2.3]{caldi1} a function $a$ satisfying the above hypothesis has a finite number of zeros in $\overline\Omega$. Notice that in such we can allow degeneracy also at some point of its boundary. To the best of our knowledge, no results are known for degenerate ``double-phase" Ambrosetti-Prodi problems, namely for equations driven by differential operators like \begin{equation}\label{marc1} \operatorname{div} (|x|^\alpha \nabla u)+\operatorname{div} (|x|^\beta |\nabla u|^{p-2}\nabla u) \end{equation} or \begin{equation}\label{marc2} \operatorname{div} (|x|^\alpha \nabla u)+\operatorname{div} (|x|^\beta\log(e+|x|) |\nabla u|^{p-2}\nabla u), \end{equation} where $\alpha\neq \beta$ are positive numbers and $1