\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2006(2006), No. 104, pp. 1--23. \newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu (login: ftp)} \thanks{\copyright 2006 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2006/104\hfil Elliptic perturbations for Hammerstein equations] {Elliptic perturbations for Hammerstein equations with singular nonlinear term} \author[G. M. Coclite, M. M. Coclite\hfil EJDE-2006/104\hfilneg] {Giuseppe Maria Coclite, Mario Michele Coclite} % in alphabetical order \address{Giuseppe Maria Coclite \newline Dipartimento di Matematica, Universit\`a di Bari, via Orabona 4, 70125 Bari, Italy} \email{coclitegm@dm.uniba.it} \address{Mario Michele Coclite \newline Dipartimento di Matematica, Universit\`a di Bari, via Orabona 4, 70125 Bari, Italy} \email{coclite@dm.uniba.it} \dedicatory{In memory of Professor Aldo Cossu} \date{} \thanks{Submitted July 3, 2006. Published September 8, 2006.} \thanks{Supported by M.U.R.S.T. Italy (funds 40\%, 60\%)} \subjclass[2000]{35B25, 45E99, 45G10, 45L99, 47H14} \keywords{Hammerstein integral equations; existence of positive solutions; \hfill\break\indent singular nonlinear boundary value problems; singular elliptic perturbations} \begin{abstract} We consider a singular elliptic perturbation of a Hammerstein integral equation with singular nonlinear term at the origin. The compactness of the solutions to the perturbed problem and, hence, the existence of a positive solution for the integral equation are proved. Moreover, these results are applied to nonlinear singular homogeneous Dirichlet problems. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{remark}[theorem]{Remark} \section{Introduction} In this paper, using elliptic perturbations, we show the existence of a positive solution to the Hammerstein equation \begin{equation} u(x)=\int_\Omega K(x,y) g\big(y,u(y)\big)dy,\quad x\in\Omega,\label{e1.1} \end{equation} where $\Omega\subset \mathbb{R}^N$, $N\ge 1$, is a bounded open set with smooth boundary and $g(y,s)$, $y\in\Omega$, $s>0$, is a positive function that is bounded in a neighborhood of $+\infty$ and possibly nonsmooth as $s\to 0^+$, in particular we do not exclude that $$ \liminf_{s\to 0^+}g(y,s)=0;\quad \limsup_{s\to 0^+}g(y,s)=+\infty. $$ Moreover, we do not assume anything about the existence of super or sub solutions to \eqref{e1.1}. The literature on Hammerstein equations with integrand depending on the reciprocal of the solution is rather limited, nevertheless they arise, more or less directly, in a variety of settings: semilinear boundary value problems with a nonlinear term depending on the reciprocal of the solution, see \cite{c2,c3,c4,c8,f2,g2} and Theorem \ref{thm2} in the following section; mathematical models of signal theory, see \cite{n1}; ecological models, see \cite[pg. 103-104]{w1}; Boussinesq's equation in filtration theory, see \cite{k1}. In literature some existence results for \eqref{e1.1} are already present (see \cite{c1,c5,c6,c7,k2}. In \cite{c7,k2}, the solutions are obtained via the perturbed problem \begin{equation} u_\varepsilon (x)=\int_\Omega K(x,y) g\big(y,\varepsilon+u_\varepsilon (y)\big)dy,\quad u_\varepsilon\in L^1(\Omega).\label{e1.2} \end{equation} The argument of this paper consists in the approximation of \eqref{e1.1} with the following elliptic integro-differential problem \begin{equation} \begin{gathered} -\varepsilon^\alpha\Delta u_\varepsilon(x)+u_\varepsilon(x)= \int_\Omega K(x,y)g\big(y,\varepsilon+u_\varepsilon(y)\big)dy\quad x\in \Omega,\\ u_\varepsilon(x)\ge 0 \quad x\in \Omega,\\ u_\varepsilon(x)=0 \quad x\in\partial\Omega, \end{gathered} \label{e1.3} \end{equation} where $\alpha>0$. The elliptic perturbations \eqref{e1.3} are interesting from both the mathematical and the physical point of view. Indeed, the solutions to \eqref{e1.3} belong to $W^{2,q}(\Omega)\cap W^{1,q}_0(\Omega)$ on the other hand the ones of \eqref{e1.2} are merely in $L^1(\Omega)$. The convergence of the solutions of the approximated problems \eqref{e1.3} to one of the integral problem \eqref{e1.1} makes easier the implementation of robust numerical schemes. In the fluidodynamic interpretation of \eqref{e1.1} in filtration theory (see \cite{k1}) the perturbation $-\varepsilon^\alpha\Delta u_\varepsilon$ represents a small viscosity. This approach to the existence of solutions for \eqref{e1.1} has been used extensively in the last years in various frameworks, in particular it gives physically meaningful solutions to Conservation Laws (see e.g. \cite{b1}). Let us be more precise regarding our results. We prove that there exist an infinitesimal sequence $({\varepsilon_k})_{{k\in\mathbb{N}}}$ and a nontrivial solution $u_0$ to \eqref{e1.1} such that $$ \lim_k\int_\Omega\eta(x) \big| u_0(x)-u_{{\varepsilon_k}}(x)\big|dx=0, $$ where $u_{{\varepsilon_k}}\in W^{2,q}(\Omega)\cap W^{1,q}_0(\Omega)$ solves \eqref{e1.3} with ${\varepsilon_k}$ instead of $\varepsilon$, $\eta(x)$ is a positive function depending on $K(x,y)$ (see the assumption (K2) in the following section), and the exponent $q$ depends on the regularity of $K(x,y)$ (see (K1). Moreover, we prove that $$\int_\Omega\eta(x)u_0(x)dx<+\infty$$ and we give an estimate on the first and second derivatives of the solutions to \eqref{e1.3} as $\varepsilon\to 0$. Finally, we consider the particular case in which $K(x,y)$ is the Green's function of $-\Delta$ on $\Omega$. We prove an existence result for homogeneous semilinear Dirichlet problems with integrand depending on the reciprocal of the solution, our result is a bit more general than the ones present in the literature, see for example \cite{c2,c3,c4,c8,f2,g2}. The starting points of our analysis are the estimates for the solutions of singular linear elliptic perturbations proved by the Huet [15] and Friedman \cite{f1}. The paper is organized as follows. Section 2 is dedicated to the assumptions and results. In Section 3 we prove the existence principle for the integral equation \eqref{e1.1}. In Section 4 we apply that result to semilinear homogeneous Dirichlet problem for $-\Delta$ with singular nonlinear term in the origin. Finally, in the appendix some convergence results are present. \vskip1truecm \section{Assumptions and results} Let us list the notation used in this paper. $$ \mathbb{R}_+:=[0,+\infty [;\quad \mathbb{R}^*_+:=]0,+\infty [; \quad \mathbb{N}^*:=\mathbb{N}\setminus\{0\}. $$ Let $ E\subset\mathbb{R}^k, k\ge 1$, be a measurable set (we will consider only measurable sets). $| E|$ is the measure of $E$, $\chi_{E}$ is the characteristic map of $E$ and $|\cdot|_{{\rho,E}}, 1\le\rho\le\infty$, is the $L^\rho(E)$ norm. $ L^\rho_+(E)$ is the cone of all $\phi\in L^\rho(E), \phi\ge0$ almost everywhere in $E$ and $L^\rho_+(\theta,E), \theta$ measurable, is the cone of all measurable $\phi, \phi\ge0$ almost everywhere in $E$, such that $\theta\phi\in L^\rho(E)$. $W^{{1\over\rho},\rho}(E)$ is the space of the maps $\phi\in L^\rho(E)$ such that $$ \int_{E\times E} {{| \phi(x)-\phi(y)|^\rho}\over {| x-y|^{k+1}}}dx\,dy<+\infty. $$ Let $u, v$ be two maps, $u\le v$ is the set of all points $x\in\Omega$ such that $u(x)\le v(x)$. Analogously, we define $uv$. We continue with the assumptions on the nonlinear term $g(y,s)$ and the kernel $K(x,y)$. Let $g:\Omega \times \mathbb{R}^*_+ \to \mathbb{R}$ be a positive Carath\`eodory function (namely $ g(\cdot ,s)$ is measurable in $\Omega$ for each $ s>0$; $g(y,\cdot)$ is continuous in $\mathbb{R}^*_+$ for almost every $ y\in\Omega$). \begin{itemize} \item[(G1)] There exist $\phi_0\in L^r(\Omega)$, $1\le r\le+\infty$, and $p>0$ such that $$ 0\le g^*(y,s)\le{{\phi_0(y)}\over{s^p}},\quad y\in\Omega,\quad 00$ and $\Omega_0\subset\Omega$, $|\Omega_0|>0$, such that $$ \liminf_{s\to 0^+}{{g(y,s)}\over s}\ge\mu_0, $$\quad uniformly with respect to $y\in\Omega_0$. \end{itemize} Let $K(x,y)$, $(x,y)\in\Omega\times\Omega$, be a nonnegative kernel and introduce the notation $$ K(\phi):=\int_\Omega K(\cdot, y)\phi(y)dy. $$ \begin{itemize} \item[(K1)] $K\in W^{{1\over q},q}(\Omega\times\Omega)$ with $10$. Hypotheses (K1), (K2) imply $a\in L^q(\Omega)$. \end{remark} \begin{remark} \label{rmk2} \rm The condition $a>0$ a.e. in $\Omega$ is equivalent to the fact that $(\Omega_n)_{n\in\mathbb{N}^*}$ covers $\Omega$. Indeed, assuming by contradiction that $|\Omega\setminus(\cup_{n=1}^\infty\Omega_n) |>0$. Since $a>0$ a.e. in $\Omega$ and $(\Omega_n)_{n\in\mathbb{N}^*}$ is increasing we have $$ 0<\int_{(\Omega\setminus(\cup_{n=1}^\infty\Omega_n))} a(x)dx=\lim_n\int_{(\Omega\setminus\Omega_n)}a(x)dx\le \lim_n {|\Omega\setminus\Omega_n|\over n}=0, $$ that is absurd. The other implication is trivial. Finally, due to the continuity of the Lebesgue measure we have also $\lim_n |\Omega\setminus\Omega_n|=0$. \end{remark} Regarding the constant $\alpha$ (see \eqref{e1.3}) we consider only the case $\alpha=10q(p+1)$. The main results of this paper are the following. \begin{theorem} \label{thm1} If \begin{equation} \mu_0 |a^2|_{{1,\Omega_0}}>1,\label{e2.1} \end{equation} then there exists a solution $u_0\in L^1_+(\eta,\Omega)$ to \eqref{e1.1} such that $|ag(\cdot,u_0)|_{{1,\Omega}}>0$ and $$ |ag(\cdot,u_0)|_{{1,\Omega}} a(x)\le u_0(x),\quad x\in\Omega \quad \text{a.e.} $$ Moreover, there exists $({\varepsilon_k})_{{k\in\mathbb{N}}}, {\varepsilon_k}\to0$, such that $$\lim_k |\eta( u_0- u_{{\varepsilon_k}})|_{{1,\Omega}}=0,\quad u_{\varepsilon_k}\to u_0 \quad\text{a.e. in }\Omega,$$ where $u_{{\varepsilon_k}}\in W^{2,q}(\Omega)\cap W^{1,q}_0(\Omega)$ solves {\rm \eqref{e1.3}}$_{{\varepsilon_k}}$. Finally, there exist $0<\bar\varepsilon\le \frac 12 $ and a constant $\bar c>0$, independent on $\varepsilon$, such that $$ \varepsilon^\alpha\sum_{i,j=1}^N |\partial^2_{i,j}u_\varepsilon|_{{q,\Omega}}+ \varepsilon^{\alpha\over 2}\sum_{i=1}^N |\partial_i u_\varepsilon |_{{q,\Omega}}\le \bar c\varepsilon^{p+2},\quad 0<\varepsilon\le\bar\varepsilon. $$ \end{theorem} If $K(x,y)$ is the Green's function of $-\Delta $ on $\Omega$, we get an existence result for the Dirichlet problem \begin{gather*} -\Delta u=g(x,u)\quad \text{in }\Omega;\\ u=0\quad\text{on }\partial\Omega. \end{gather*} \begin{theorem} \label{thm2} Let $N\ge 2$. Assume that $g(y,s)$ satisfies {\rm (G1), (G2)} with $$ q<{N\over{N-1}};\quad q+r\le rq;\quad {{\phi_0}\over{\delta^{p^*-1}}}\in L^1(\Omega),\quad p^*=\max\{p,1\}, $$ where $\delta(x)=\mathop{\rm dist}(x,\partial\Omega),\quad x\in\mathbb{R}^N$. If \begin{equation} \mu_0 |\phi_{1}^2|_{{1,\Omega_0}}>1,\label{e2.2} \end{equation} where $\phi_{1}$ is a positive eigenfunction of the Dirichlet problem for $-\Delta$ in $\Omega$ such that $$ \phi_{1}(x)\phi_{1}(y)\le G(x,y), $$ then there exist $u_0\in W^{2,r}_{\rm loc}(\Omega)\cap C(\bar\Omega)$ and $c_2>0$ such that $c_2\delta(x)\le u_0(x)$ and \begin{equation} \begin{gathered} -\Delta u_0(x)=g\big(x,u_0(x)\big)\quad x\in \Omega,\\ u_0(x)> 0 \quad x\in \Omega, \\ u_0(x)=0 \quad x\in\partial\Omega. \end{gathered} \label{e2.3} \end{equation} Moreover, for every $\varepsilon>0$ there exists $u_\varepsilon\in W^{4,r}(\Omega)$, a solution of \begin{equation} \begin{gathered} \varepsilon^\alpha\Delta^2 u_\varepsilon(x)-\Delta u_\varepsilon(x)= g\big(x,\varepsilon+u_\varepsilon(x)\big)\quad x\in \Omega,\\ u_\varepsilon(x)> 0 \quad x\in\Omega,\\ \Delta u_\varepsilon(x)=u_\varepsilon(x)=0 \quad x\in\partial\Omega, \end{gathered} \label{e2.4} \end{equation} and $({\varepsilon_k})_{{k\in\mathbb{N}}}, {\varepsilon_k}\to 0$, such that $u_{{\varepsilon_k}}\to u_0\>\>{ in}\>\>W^{2,r}_{\rm loc}(\Omega) \cap L^q(\Omega)$. \end{theorem} In light of Lemma \ref{lem4.1} below, $G(x,y)$ satisfies (K1), (K2), (K3) with $$ q<{N\over {N-1}};\quad a(x)={1\over{\sqrt{c_{1}}}}\delta(x);\quad \eta(x)={1\over{c_{1}\sqrt{c_{1}}}}, $$ hence the integral equation associated with \eqref{e2.3} satisfies the same hypotheses of \eqref{e1.1}. Since $\delta$ is equivalent to each positive eigenfunction of the Dirichlet problem for $-\Delta$ in $\Omega$ (see \cite{c2}), \eqref{e2.2} coincides with \eqref{e2.1} when $K=G$. \section{Proof of Theorem \ref{thm1}} In the following statements and proofs we write ``cost" for positive constants independent of $\varepsilon$. The first step of our analysis consists in the existence of solutions for \eqref{e1.3}. Thanks to (G1), (K1), $$ K(g^*(\cdot,\varepsilon))=\int_\Omega K(\cdot,y)g^*(y,\varepsilon)dy \in L^q(\Omega),\quad \varepsilon>0; $$ therefore, $K(g(\cdot,\varepsilon+u))\in L^q(\Omega),\quad \varepsilon>0$, $u\in L^q_+(\Omega)$. Due to \cite[Theorem 9.15]{g1}, for each $\varepsilon>0$ and $ u \in L^q_+(\Omega)$, there exists a unique $\Phi_\varepsilon(u)\in W^{2,q}(\Omega)\cap W^{1,q}_0(\Omega)$ such that \begin{gather*} -\varepsilon^\alpha\Delta\Phi_\varepsilon(u)+\Phi_\varepsilon(u)= K(g(\cdot,\varepsilon+u)) \quad\text{in }\Omega,\\ \Phi_\varepsilon(u)\ge 0 \quad\text{in }\Omega,\\ \Phi_\varepsilon(u)=0 \quad\text{on }\partial\Omega. \end{gather*} Analogously, there exists a unique $U_\varepsilon\in W^{2,q}(\Omega)\cap W^{1,q}_0(\Omega)$ such that \begin{gather*} -\varepsilon^\alpha\Delta U_\varepsilon+U_\varepsilon=K(g^*(\cdot,\varepsilon)) \quad\text{in }\Omega, \\ U_\varepsilon\ge 0 \quad\text{in }\Omega,\\ U_\varepsilon=0 \quad\text{on } \partial\Omega. \end{gather*} Since $00$. \begin{itemize} \item[(i)] $\Phi_\varepsilon $ is continuous in the sense that for every $(u_n)_{n\in\mathbb{N}}$ and $\bar u$ in $L^q_+(\Omega)$, $$ u_n\to \bar u\>\>in\,L^q(\Omega)\;\Rightarrow\; \Phi_\varepsilon(u_n)\,\to\,\Phi_\varepsilon(\bar u) \quad \text{in } W^{2,q}(\Omega). $$ \item[(ii)] $\Phi_\varepsilon(L^q_+(\Omega))$ is compact in $L^q(\Omega)$. \end{itemize} \end{lemma} \begin{proof} (i) Let $(u_n)_{n\in\mathbb{N}}$ and $\bar u$ in $L^q_+(\Omega)$ be such that $u_n\to \bar u$ in $L^q(\Omega)$. By (G1), $$ 0\le g(\cdot,\varepsilon+u_n)\le g^*(\cdot,\varepsilon )\in L^r(\Omega)\subset L^{q'}(\Omega),\quad n\in\mathbb{N}. $$ Due to the continuity of the Nemytskii operator $u\in L^q(\Omega)\mapsto g(\cdot, \varepsilon+u)\in L^{q'}(\Omega):$ $$ g(\cdot,\varepsilon+u_n)\to g(\cdot,\varepsilon+\bar u )\quad\text{in }L^{q'}(\Omega). $$ Then \begin{equation} K(g(\cdot,\varepsilon+u_n))\to K(g(\cdot,\varepsilon+\bar u))\quad \text{in } L^q(\Omega).\label{e3.1} \end{equation} Since $\Phi_\varepsilon(u_n)-\Phi_\varepsilon(\bar u)\in W^{2,q}(\Omega) \cap W^{1,q}_0(\Omega)$ and $$ -\varepsilon^\alpha\Delta\big[\Phi_\varepsilon(u_n)-\Phi_\varepsilon(\bar u)\big]+ \big[\Phi_\varepsilon(u_n)-\Phi_\varepsilon(\bar u)\big]= K(g(\cdot,\varepsilon+u_n))-K(g(\cdot,\varepsilon+\bar u)), $$ employing \cite[Lemma 9.17]{g1}, there exists $c_\varepsilon>0$ independent on $u_n$ and $\bar u$, such that $$ \Vert\Phi_\varepsilon(u_n)-\Phi_\varepsilon(\bar u)\Vert_{{W^{2,q}(\Omega)}}\le c_\varepsilon |K(g(\cdot,\varepsilon+u_n))-K(g(\cdot,\varepsilon+\bar u))|_{{q,\Omega}}. $$ Hence (i) follows from \eqref{e3.1}. \smallskip \noindent (ii) Let $(u_n)_{n\in \mathbb{N}}$, $u_n\in L^q_+(\Omega)$ be bounded. We prove that $(\Phi_\varepsilon(u_n))_{n\in \mathbb{N}}$ has a converging subsequence in $ L^q(\Omega)$. Due to (G1) and ((K1), $(K(g(\cdot,\varepsilon+u_n)))_{n\in\mathbb{N}}$ is bounded in $L^q(\Omega)$. Hence $(\Phi_\varepsilon(u_n))_{n\in\mathbb{N}}$ is bounded in $W^{1,q}_0(\Omega)$ (see \cite[Lemma 9.17]{g1}). Using $W^{1,q}_0(\Omega)\hookrightarrow$ $\hookrightarrow L^q(\Omega)$, $(\Phi_\varepsilon(u_n))_{n\in\mathbb{N}}$ has a converging subsequence in $L^q(\Omega)$. The lemma is proved. \end{proof} \begin{corollary} \label{coro3.2} For each $\varepsilon>0$ there exists $u_\varepsilon \in S_\varepsilon\subset W^{2,q}(\Omega)$, such that $u_\varepsilon=\Phi_\varepsilon(u_\varepsilon)$, namely \begin{gather*} -\varepsilon^\alpha\Delta u_\varepsilon+u_\varepsilon=K(g(\cdot,\varepsilon+u_\varepsilon))\quad \quad\text{in } \Omega,\\ u_\varepsilon\ge 0\quad\text{in } \Omega, \\ u_\varepsilon=0\quad \text{on } \partial\Omega. \end{gather*} \end{corollary} The claim of the above corollary follows directly from the Schauder theorem. The following two lemmas play a key role in our argument. \begin{lemma}[{\cite[Proposition 2.1]{h1}}] \label{lem3.3} Let $1<\rho<\infty$ and $\lambda_{1}$ be the first eigenvalue of the Dirichlet problem for $-\Delta$ on $\Omega$. For every $0<\varepsilon<{1\over{\lambda_{1}}}$ and $\omega\in W^{2,\rho}(\Omega)\cap W^{1,\rho}_0(\Omega)$ we have that $$ \varepsilon^\alpha{\Vert\omega\Vert_{{W^{2,\rho}(\Omega)}}}+\varepsilon^{\alpha\over 2} {\Vert\omega\Vert_{{W^{1,\rho}(\Omega)}}}+{|\omega|_{{\rho,\Omega}}} \le\mathop{\rm const}{|-\varepsilon^\alpha\Delta\omega+\omega|_{{\rho,\Omega}}}. $$ Moreover, if $\omega_\varepsilon\in W^{2,\rho}(\Omega)$ solves \begin{gather*} -\varepsilon^\alpha\Delta\omega_\varepsilon+\omega_\varepsilon=h_\varepsilon \quad\text{in } \Omega,\\ \omega_\varepsilon=0 \quad \text{on } \partial\Omega , \end{gather*} with $h_\varepsilon\in L^\rho(\Omega)$ converging as $\varepsilon\to 0$ in $L^\rho(\Omega)$, there results $$ \lim_{\varepsilon\to 0}|\omega_\varepsilon-h_\varepsilon|_{{\rho,\Omega}}=0;\quad \lim_{\varepsilon\to 0}\varepsilon^\alpha \Vert\omega_\varepsilon\Vert_{{W^{2,\rho}(\Omega)}}=0. $$ \end{lemma} \begin{lemma} \label{lem3.4} For each $\varepsilon>0$ there exists a unique $K_\varepsilon\in W^{2,q}(\Omega\times\Omega)\cap W^{1,q}_0(\Omega\times\Omega)$ such that \begin{gather*} -\varepsilon^{6q(p+1)}\Delta K_\varepsilon+ K_\varepsilon= K \quad\text{in }\Omega\times\Omega,\\ K_\varepsilon\ge 0\quad\text{in }\Omega\times\Omega,\\r K_\varepsilon=0 \quad\text{on } \partial(\Omega\times\Omega). \end{gather*} Moreover, \begin{itemize} \item[(i)] $\varepsilon^{6q(p+1)} {\Vert K_\varepsilon\Vert_{{W^{2,q}(\Omega\times\Omega)}}}+ \varepsilon^{3q(p+1)} {\Vert K_\varepsilon\Vert_{{W^{1,q}(\Omega\times\Omega)}}}+ {| K_\varepsilon|_{{q,\Omega\times\Omega}}}\\ \le\mathop{\rm const} {| K|_{{q,\Omega\times\Omega}}}$. \item[(ii)] There exists $\varepsilon_0>0$ such that $| K_\varepsilon-K|_{{q,\Omega\times\Omega}}\\ \le \mathop{\rm const} (\varepsilon^{6q(p+1)})^{1\over{3q}} \Vert K\Vert_{W^{{1\over q}, q}(\Omega\times\Omega)}$, $0<\varepsilon<\varepsilon_0$. \end{itemize} \end{lemma} \begin{proof} Part (i) follows from the previous lemma, and (ii) follows from \cite[Theorem 1.2]{f1}. \end{proof} For short, introduce the notation $$ g_\varepsilon:=g(\cdot,\varepsilon+u_\varepsilon);\quad \bar\varepsilon:=\min\{\varepsilon_0,\frac 12 \}. $$ For the rest of this article, we assume that $0<\varepsilon\le\bar\varepsilon$. The second step of our argument consists in the following estimates. \begin{lemma} \label{lem3.5} The following results hold: \begin{itemize} \item[(a)] $| K(g_\varepsilon)|_{{q,\Omega}}\le\mathop{\rm const}\varepsilon^{-p}$. (It suffices to assume $ K\in L^q(\Omega\times\Omega)$.) \item[(b)] $| K_\varepsilon (g_\varepsilon)|_{{q,\Omega}}\le\mathop{\rm const}\varepsilon^{-p}$. \item[(c)] $\Vert K_\varepsilon (g_\varepsilon)\Vert_{{W^{1,q}(\Omega)}}\le\mathop{\rm const} \varepsilon^{-p-3q(p+1)}$. \item[(d)] $\Vert K_\varepsilon (g_\varepsilon)\Vert_{{W^{2,q}(\Omega)}}\le\mathop{\rm const} \varepsilon^{-p-6q(p+1)}$. \item[(e)] $| K(g_\varepsilon)-K_\varepsilon(g_\varepsilon)|_{{q,\Omega}}\le\mathop{\rm const} \varepsilon^{p+2}$. \item[(f)] $| u_\varepsilon|_{{q,\Omega}}\le\mathop{\rm const}\varepsilon^{-p}$. \item[(g)] $\Vert u_\varepsilon\Vert_{{W^{1,q}(\Omega)}} \le \mathop{\rm const}\varepsilon^{-p-5q(p+1)}$. \end{itemize} \end{lemma} \begin{proof} We begin by proving \begin{equation} | K(g_\varepsilon)|_{{q,\Omega}}\le\mathop{\rm const}\varepsilon^{-p} | K|_{{q,\Omega\times\Omega}},\label{e3.2} \end{equation} which implies (a). Since $q+r\le qr$ is equivalent to $q'\le r$, thanks to (G1), $\phi_0, g^*(\cdot,\frac 12 )\in L^{q'}(\Omega)$. Define $X=(u_\varepsilon\le\frac 12 )$, \begin{align*} &| K(g_\varepsilon)|_{{q,\Omega}}\\ &\le| K(g_\varepsilon\chi_{{X}})|_{{q,\Omega}}+ | K(g_\varepsilon\chi_{{\Omega\setminus X}})|_{{q,\Omega}}\\ &\le\Big(\int_\Omega\big(\int_\Omega K(x,y) {{\phi_0(y)}\over{(\varepsilon+u_\varepsilon(y))^p}}dy\big)^qdx\Big)^{1\over q}+ \Big(\int_\Omega\big(\int_\Omega K(x,y) g^*\big(y,\frac 12 \big)dy\big)^qdx\Big)^{1\over q}\\ &\le {1\over{\varepsilon^p}}\Big(\int_\Omega\big(\int_\Omega K(x,y)^qdy\big)|\phi_0|_{{q',\Omega}}^qdx \Big)^{1\over q}+ \Big(\int_\Omega\big(\int_\Omega K(x,y)^qdy\big) \Big|g^*\big(\cdot,\frac 12 \big)\Big|_{{q',\Omega}}^qdx \Big)^{1\over q}\\ &\le \Big({{|\phi_0|_{{q',\Omega}}}\over{\varepsilon^p}}+ \Big| g^*(\cdot,\frac 12 )\Big|_{{q',\Omega}}\Big) | K|_{{q,\Omega\times\Omega}}. \end{align*} Hence \eqref{e3.2} is proved. Analogously we can prove $$ | K_\varepsilon (g_\varepsilon)|_{{q,\Omega}}\le\mathop{\rm const}\varepsilon^{-p} | K_\varepsilon|_{{q,\Omega\times\Omega}}. $$ Then employing \eqref{e3.4}(i), we get (b). Since $$ \partial_i K_\varepsilon(g_\varepsilon)(x)=\int_\Omega\partial_{x_i} K_\varepsilon(x,y)g_\varepsilon(y)dy;\quad \partial^2_{i,j} K_\varepsilon(g_\varepsilon)(x)=\int_\Omega\partial^2_{x_i,x_j} K_\varepsilon(x,y)g_\varepsilon(y)dy, $$ arguing in the same way we have \begin{gather*} |\partial_i K_\varepsilon (g_\varepsilon)|_{{q,\Omega}}\le\mathop{\rm const} \varepsilon^{-p} |\partial_{x_i} K_\varepsilon|_{{q,\Omega\times\Omega}};\\ |\partial^2_{i,j} K_\varepsilon (g_\varepsilon)|_{{q,\Omega}}\le\mathop{\rm const}\varepsilon^{-p} |\partial^2_{x_i,x_j} K_\varepsilon|_{{q,\Omega\times\Omega}}. \end{gather*} Employing again \eqref{e3.4}(ii) we get (c), (d). Using the same argument of \eqref{e3.2}, $$ | K(g_\varepsilon)-K_\varepsilon(g_\varepsilon)|_{{q,\Omega}}\le\mathop{\rm const}\varepsilon^{-p} | K-K_\varepsilon|_{{q,\Omega\times\Omega}}, $$ by \eqref{e3.4}(ii), $$ | K(g_\varepsilon)-K_\varepsilon(g_\varepsilon)|_{{q,\Omega}}\le\mathop{\rm const}\varepsilon^{p+2} \Vert K\Vert_{{W^{{1\over q},q} (\Omega\times\Omega)}}, $$ that is (e). Applying Lemma \ref{lem3.3} to $u_\varepsilon$ in light of \eqref{e3.2} $$ \varepsilon^{5q(p+1)}\Vert u_\varepsilon\Vert_{{W^{1,q}(\Omega)}}+ |u_\varepsilon|_{{q,\Omega}}\le\mathop{\rm const} |K(g_\varepsilon)|_{{q,\Omega}}\le\mathop{\rm const} {\varepsilon^{-p}}, $$ from which (f) and (g) follow. \end{proof} \begin{lemma} \label{lem3.6} The following estimate holds $$ \varepsilon^\alpha\sum_{i,j=1}^N |\partial_{i,j}^2u_\varepsilon|_{{q,\Omega}}+ \varepsilon^{\alpha\over 2}\sum_{i=1}^N |\partial_i u_\varepsilon |_{{q,\Omega}}+ | u_\varepsilon-K(g_\varepsilon)|_{{q,\Omega}}\le\bar c\varepsilon^{p+2}, $$ for some constant $\bar c>0$ independent of $\varepsilon$. \end{lemma} \begin{proof} We begin by proving that $K_\varepsilon(g_\varepsilon)\in W^{2,q}(\Omega)\cap W^{1,q}_0(\Omega)$. The fact $K_\varepsilon(g_\varepsilon)\in W^{2,q}(\Omega)$ follows from \eqref{e3.5}(d). Due to Lemma \ref{lem3.4}, $K_\varepsilon\in W_0^{1,q}(\Omega\times\Omega)$. By definition of $W_0^{1,q}(\Omega\times\Omega)$ there exists $(\Phi_n)_{n}, \Phi_n\in C_0^\infty(\Omega\times\Omega)$ such that \begin{equation} \lim_n\Vert K_\varepsilon-\Phi_n\Vert_{{W^{1,q}(\Omega\times\Omega)}}=0. \label{e3.3} \end{equation} Denote $$ \Phi_n(g_\varepsilon)(x)=\int_\Omega\Phi_n(x,y)g_\varepsilon(y)dy, \quad x\in\Omega. $$ Since $\partial\Omega\times\Omega\subset\partial(\Omega\times\Omega)$ we have that $\Phi_n(g_\varepsilon)\in C_0^\infty(\Omega)$. Using the H\"older Inequality, \begin{align*} \Vert K_\varepsilon(g_\varepsilon)-\Phi_n(g_\varepsilon)\Vert_{{W^{1,q}(\Omega)}} &= \sum_{i=1}^N |\partial_i K_\varepsilon(g_\varepsilon)- \partial_i\Phi_n(g_\varepsilon)|_{{q,\Omega}}+ |K_\varepsilon(g_\varepsilon)-\Phi_n(g_\varepsilon)|_{{q,\Omega}}\\ &\le\Big(\sum_{i=1}^N |\partial_{x_i} K_\varepsilon-\partial_{x_i}\Phi_n|_{{q,\Omega}}+ |K_\varepsilon-\Phi_n|_{{q,\Omega}}\Big) |g_\varepsilon|_{{_{q',\Omega}}}\\ &= \Vert K_\varepsilon-\Phi_n\Vert_{{W^{1,q}(\Omega)}} |g_\varepsilon|_{{_{q',\Omega}}}. \end{align*} Employing \eqref{e3.3}, $$ \lim_n \Vert K_\varepsilon(g_\varepsilon)-\Phi_n(g_\varepsilon)\Vert_{{W^{1,q}(\Omega)}} =0, $$ then by definition, $K_\varepsilon(g_\varepsilon)\in W^{1,q}_0(\Omega)$. We continue by observing that $$ -\varepsilon^\alpha\Delta(u_\varepsilon-K_\varepsilon(g_\varepsilon))+(u_\varepsilon-K_\varepsilon(g_\varepsilon))= K(g_\varepsilon)-K_\varepsilon(g_\varepsilon)+\varepsilon^\alpha\Delta K_\varepsilon(g_\varepsilon). $$ From Lemma \ref{lem3.3}, \begin{align*} &\varepsilon^\alpha\Vert u_\varepsilon-K_\varepsilon(g_\varepsilon)\Vert_{{W^{2,q}(\Omega)}}+ \varepsilon^{\alpha\over 2}\Vert u_\varepsilon-K_\varepsilon(g_\varepsilon)\Vert_{{W^{1,q}(\Omega)}}+ |u_\varepsilon-K_\varepsilon(g_\varepsilon)|_{{q,\Omega}}\\ &\le \mathop{\rm const} |K(g_\varepsilon)-K_\varepsilon(g_\varepsilon)+ \varepsilon^\alpha\Delta K_\varepsilon(g_\varepsilon)|_{{q,\Omega}}; \end{align*} hence \begin{align*} L_\varepsilon&:=\varepsilon^\alpha\Vert u_\varepsilon\Vert_{{W^{2,q}(\Omega)}}+ \varepsilon^{\alpha\over 2}\Vert u_\varepsilon\Vert_{{W^{1,q}(\Omega)}}+ | u_\varepsilon-K(g_\varepsilon)|_{{q,\Omega}}\\ &\le\mathop{\rm const}\Big(|K(g_\varepsilon)-K_\varepsilon(g_\varepsilon)|_{{q,\Omega}}+ \varepsilon^\alpha\Vert K_\varepsilon(g_\varepsilon)\Vert_{{W^{2,q}(\Omega)}}+ \varepsilon^{\alpha\over 2}\Vert K_\varepsilon(g_\varepsilon)\Vert_{{W^{1,q}(\Omega)}}\Big). \end{align*} Using Lemma \ref{lem3.5}, $$ L_\varepsilon\le\mathop{\rm const}\big(\varepsilon^{p+2}+ \varepsilon^{4q(p+1)-p}+ \varepsilon^{2q(p+1)-p}\big). $$ Since $p+2< 2q(p+1)-p<4q(p+1)-p$, we have $L_\varepsilon\le\mathop{\rm const}\varepsilon^{p+2}$. This gives the claim. \end{proof} \begin{lemma} \label{lem3.7} The sequence $(ag_\varepsilon)_{0<\varepsilon<\bar\varepsilon}$ is equiabsolutely continuous, more precisely for each $E\subset\subset\Omega$, $$ | ag_\varepsilon|_{{1,E}}\le T(E),\quad 0<\varepsilon\le\bar\varepsilon, $$ where \begin{gather*} T(E)={A(E)+C(E)+\sqrt{B(E)+C(E)}};\quad A(E)=\Big| ag^* (\cdot ,\frac 12 )\Big|_{{1,E}}; \\ B(E)=\bar c |\phi_0|_{{q',E}}; \quad C(E)=\Big|{{\phi_0}\over{a^{p^*-1}}}\Big|^{1\over p^*}_{{1,E}}. \end{gather*} \end{lemma} \begin{proof} Define $X=E\cap(u_\varepsilon\le\frac 12 )$. Multiplying \eqref{e1.3} by $g_\varepsilon$ and integrating on $X:$ \begin{equation} -\varepsilon^\alpha\int_X(\Delta u_\varepsilon) g_\varepsilon dx+\int_Xu_\varepsilon g_\varepsilon dx= \int_X K(g_\varepsilon)g_\varepsilon dx.\label{e3.4} \end{equation} We continue by estimating separately the three terms. By (K1), $q'\le r$, hence $\phi_0\in L^{q'}(\Omega)$ (see (G1)) so $$ -\varepsilon^\alpha\int_X \Delta u_\varepsilon g_\varepsilon dx\le\varepsilon^{\alpha-p} \sum_{i,j=1}^N\int_X |\partial_{i,j}^2 u_\varepsilon|\phi_0dx \le\varepsilon^{\alpha-p}\sum_{i,j=1}^N |\partial_{i,j}^2u_\varepsilon|_{{q,X}} \cdot |\phi_0|_{{q',X}}. $$ Since $\varepsilon\le \frac 12 $, from Lemma \ref{lem3.6}, \begin{equation} -\varepsilon^\alpha\int_X \Delta u_\varepsilon g_\varepsilon dx\le \bar c\varepsilon^2 |\phi_0|_{{q',E}}\le {{\bar c}\over 4} |\phi_0|_{{q',E}}\le B(E).\label{e3.5} \end{equation} We distinguish two cases. If $p\le 1$, $$ \int_X u_\varepsilon g_\varepsilon dx\le\int_X u_\varepsilon^{1-p}\phi_0dx\le {1\over{2^{1-p}}}|\phi_0|_{{1,X}}\le |\phi_0|_{{1,E}}.$$ If $p> 1$, $$ \int_X u_\varepsilon g_\varepsilon dx= \int_X u_\varepsilon g_\varepsilon^{1/p} g_\varepsilon^{1\over{p'}}dx\le \int_X{{\phi_0^{{1}\over{p}}}\over{a^{{1}\over{p'}}}} (ag_\varepsilon)^{{1}\over {p'}} dx \le\Big|{{\phi_0}\over{a^{p-1}}}\Big|^{1/p}_{{1,E}}\cdot |ag_\varepsilon|^{1\over{p'}}_{{1,X}}, $$ where $p'={p\over{p-1}}$. Therefore, \begin{equation} \int_X u_\varepsilon g_\varepsilon dx\le \begin{cases} C(E)&\text{if }p\le 1,\\ C(E)\cdot |ag_\varepsilon|^{1\over{p'}}_{{1,X}}&\text{if }p>1. \end{cases} \label{e3.6} \end{equation} Finally, from (K2), \begin{equation} \label{e3.7} \begin{aligned} \int_XK(g_\varepsilon)g_\varepsilon dx &\ge \int_{X\times\Omega}a(x)g_\varepsilon(x)a(y)g_\varepsilon(y)dx\,dy\\ &\ge | ag_\varepsilon|_{{1,\Omega}}\cdot| ag_\varepsilon|_{{1,X}}\\ &\ge |ag_\varepsilon |_{{1,E}}\big(|ag_\varepsilon |_{{1,E}}-| ag_\varepsilon|_{{1,E\setminus X}}\big)\\ & \ge|ag_\varepsilon |_{{1,E}}\big(|ag_\varepsilon |_{{1,E}}- \big|ag^*(\cdot,\frac 12 )\big|_{{1,E}}\big). \end{aligned} \end{equation} Using \eqref{e3.5}, \eqref{e3.6}, \eqref{e3.7} in \eqref{e3.4}, we obtain that $p\le 1$ implies $$ \bar c |\phi_0|_{{q',E}}+ |\phi_0|_{{1,E}}+ |ag_\varepsilon|_{{1,E}}\Big|ag^*(\cdot,\frac 12 )\Big|_{{1,E}}\ge |ag_\varepsilon|^2_{{1,E}} $$ which in turn implies $$ |ag_\varepsilon|_{{1,E}}\le A(E)+\sqrt{B(E)+C(E)}. $$ Also $p> 1$ implies $$ \bar c |\phi_0|_{{q',E}}+ \Big|{{\phi_0}\over{a^{p-1}}}\Big|^{1/p}_{{1,E}}\cdot |ag_\varepsilon|^{1\over{p'}}_{{1,E}}+ |ag_\varepsilon|_{{1,E}}\Big|ag^*(\cdot,\frac 12 )\Big|_{{1,E}}\ge |ag_\varepsilon|^2_{{1,E}}. $$ Denoting $\theta=| a g_\varepsilon|_{{1,E}}$, the previous estimate becomes $$ \theta^2\le A(E)\theta+C(E)\theta^{1\over{p'}}+B(E). $$ Then $p>1$ and $\theta\le 1$ imply $$ \theta^2\le A(E)\theta+B(E)+C(E) \;\Rightarrow\;\theta\le A(E)+\sqrt{B(E)+C(E)}. $$ Also $p>1$ and $\theta>1$ imply $$ \theta^2\le (A(E)+C(E))\theta+B(E) \;\Rightarrow\;\theta\le A(E)+C(E)+\sqrt{B(E)}. $$ In conclusion, for every $p$, we have $$ \theta\le A(E)+C(E)+\sqrt{B(E)+C(E)}. $$ The proof is complete. \end{proof} In light of the estimates of the previous lemmas we are now able to prove that the family $(u_\varepsilon)_{0<\varepsilon\le \bar\varepsilon}$ is compact and has a subsequence that converges to a positive solution of \eqref{e1.1}. \begin{lemma} \label{lem3.8} There exists $({\varepsilon_k})_{{k\in\mathbb{N}}}$, ${\varepsilon_k}\to0$, such that $(K(g_{{\varepsilon_k}}\chi_{{\Omega_n}}))_{{k\in\mathbb{N}}}$ converges in $L^1(\Omega_n)$, for each $n\in\mathbb{N}^*$. \end{lemma} \begin{proof} Due to the previous lemma, $(g_\varepsilon)_{{0<\varepsilon\le\bar\varepsilon}}$ is bounded in $L^1(\Omega_1)$, and by $(\mathcal{K}_3)$ there exists $(\varepsilon_{1,k})_{{k\in\mathbb{N}}}, \varepsilon_{1,k}\to 0$, such that $(K(g_{\varepsilon_{1,k}}\chi_{{\Omega_1}}))_{{k\in\mathbb{N}}}$ is converging in $L^1(\Omega_1)$. Iterating this argument, for each $n\in\mathbb{N}$ there exist $(\varepsilon_{i,k})_{{k\in\mathbb{N}}}, 1\le i\le n$, tending to 0 with $(\varepsilon_{j+1,k})_{{k\in\mathbb{N}}}$ subsequence of $(\varepsilon_{j,k})_{{k\in\mathbb{N}}}$ and $(K(g_{\varepsilon_{j,k}}\chi_{{\Omega_i}}))_{{k\in\mathbb{N}}}$ converging in $L^1(\Omega_j), 1\le j\le n$. Hence, by induction there exists $(\varepsilon_{i,k})_{k\in\mathbb{N}}$ playing the same game. $(\varepsilon_{k,k})_{k\in\mathbb{N}^*}$ is a subsequence of every $(\varepsilon_{i,k})_{{k\in\mathbb{N}}}$, hence it fulfills the claim. \end{proof} Thanks to the previous lemma we can define $$ v_n:=\begin{cases} \lim_k K(g_{{\varepsilon_k}}\chi_{{\Omega_n}}), & \text{in } \Omega_n,\\ 0, &\text{in } \Omega\setminus\Omega_n. \end{cases} $$ From Lemma \ref{lem3.8}, $v_n\in L^1(\Omega)$, and by construction $(v_n)_{n\in\mathbb{N}^*}$ is increasing, so $$ u_0:=\lim_n v_n=\sup_n v_n. $$ \begin{lemma} \label{lem3.9} $u_0$ satisfies the following conditions: \begin{itemize} \item[(a)] $u_0\in L^1_+(\eta,\Omega)$ and $|\eta u_0|_{{1,\Omega}}\le T(\Omega)$. \item[(b)] $\lim_n|\eta (u_0-v_n)|_{{1,\Omega}}=0$. \item[(c)] For all $n\in\mathbb{N}^*$, $\lim_k \Big|{\eta\over{1+\eta}} (u_0-K(g_{{\varepsilon_k}}))\Big|_{{1,\Omega_n}}=0$. \item[(d)] For all $ n\in\mathbb{N}^*$, $\lim_k \Big|{\eta\over{1+\eta}} (u_0-u_{{\varepsilon_k}})\Big|_{{1,\Omega_n}}=0$. \item[(e)] Passing to a subsequence $K(g_{{\varepsilon_k}})\to u_0$, $u_{{\varepsilon_k}}\to u_0$, a.e. in $\Omega$. \end{itemize} \end{lemma} \begin{proof} (a) By Lemma \ref{lem3.7} and (K2), $$ T(\Omega)\ge |ag_{\varepsilon_k}|_{{1,\Omega}}\ge\int_{\Omega_n}g_{\varepsilon_k}(y)dy \int_\Omega K(x,y)\eta(x)dx\ge |\eta K(g_{\varepsilon_k}\chi_{{\Omega_n}})|_{{1,\Omega_n}}. $$ Sending $n, k\to+\infty$, $$ T(\Omega)\ge\lim_n\lim_k |\eta K(g_{\varepsilon_k}\chi_{{\Omega_n}})|_{{1,\Omega_n}}= \lim_n |\eta v_n|_{{1,\Omega_n}}= \lim_n |\eta v_n|_{{1,\Omega}}=|\eta u_0|_{{1,\Omega}}. $$ Part (b) is a direct consequence of the definition of $u_0$, (a) and the Dominate Convergence Theorem. \noindent(c) Let $m\ge n>0$ be integer numbers. Observe that \begin{align*} &\Big|{\eta\over{1+\eta}} (u_0- K(g_{\varepsilon_k}))\Big|_{{1,\Omega_n}}\\ &\le |\eta(u_0- v_m)|_{{1,\Omega_n}}+ | v_m- K(g_{\varepsilon_k}\chi_{{\Omega_m}})|_{{1,\Omega_n}}+ |\eta K(g_{\varepsilon_k}\chi_{{\Omega\setminus\Omega_m}})|_{{1,\Omega}} \end{align*} and $\lim_k|v_m- K(g_{\varepsilon_k}\chi_{{\Omega_m}})|_{{1,\Omega_n}}=0$. By (K2) and Lemma \ref{lem3.7}, $$ |\eta K(g_{\varepsilon_k}\chi_{{\Omega\setminus\Omega_m}})|_{{1,\Omega}}\le |ag_{\varepsilon_k}|_{{1,\Omega\setminus\Omega_m}}\le T(\Omega\setminus\Omega_m). $$ Hence $$ \limsup_k \Big|{\eta\over{1+\eta}} (u_0- K(g_{\varepsilon_k}))\Big|_{{1,\Omega_n}}\le |\eta(u_0- v_m)|_{{1,\Omega_n}}+T(\Omega\setminus\Omega_m). $$ Since $|\Omega\setminus\Omega_m|\to 0$, using the absolute continuity of the integrals in $T(\cdot)$ we have that $T(\Omega\setminus\Omega_m)\to 0$. Hence (b) implies (c). \noindent(d) Due to Lemma \ref{lem3.6} and (K2), \begin{align*} \Big|{\eta\over{1+\eta}}(u_0- u_{\varepsilon_k})\Big|_{{1,\Omega_n}} &\le \Big|{\eta\over{1+\eta}}(u_0- K(g_{\varepsilon_k}))\Big|_{{1,\Omega_n}}+ |\eta(K(g_{\varepsilon_k})- u_{\varepsilon_k})|_{{1,\Omega_n}}\\ &\le \Big|{\eta\over{1+\eta}}(u_0- K(g_{\varepsilon_k}))\Big|_{{1,\Omega_n}}+ |\eta|_{{q',\Omega}} |K(g_{\varepsilon_k})-u_{\varepsilon_k}|_{{q,\Omega}}\\ &\le \Big|{\eta\over{1+\eta}}(u_0- K(g_{\varepsilon_k}))\Big|_{{1,\Omega_n}}+ |\eta|_{{_{q',\Omega}}} \bar c{\varepsilon_k}^{p+2}, \end{align*} using (c) we have (d). Part e) is a consequence of (c), (d) and of the positivity of the map $\eta$ a.e. in $\Omega$; see (K2). \end{proof} In the proof of Lemma \ref{lem3.11} we will use the following convergence theorem that will be proved in the appendix. \begin{lemma} \label{lem3.10} Let $f_{k}\in L^1(\Omega)$, $\phi_{k}\in L^\infty(\Omega)$, $k\in\mathbb{N}$, and $\Omega\subset\mathbb{R}^N$. If $|\Omega|<\infty$, $(f_{k})_{{k\in\mathbb{N}}}$ is bounded in $L^1(\Omega)$ and equiabsolutely continuous, $(\phi_{k})_{{k\in\mathbb{N}}}$ is bounded in $L^\infty(\Omega)$, and converging in measure to $\phi\in L^\infty(\Omega)$, then \begin{itemize} \item[(i)] $\lim_k| f_{k}\phi_{k}-f_{k}\phi|_{{1,\Omega}}=0$ \item[(ii)] $$ \limsup_k| f_{k}\phi_{k}|_{{1,\Omega}}= \limsup_k| f_{k}\phi |_{{1,\Omega}};\quad \liminf_k| f_{k}\phi_{k}|_{{1,\Omega}}= \liminf_k| f_{k}\phi |_{{1,\Omega}}. $$ \end{itemize} \end{lemma} \begin{lemma} \label{lem3.11} Let $n\in\mathbb{N}^*$ and $L>0$. Then $$ \limsup_k| a g_{{\varepsilon_k}}|_{{1,\Omega_n\cap (u_0\le L)}}\le nL(1+L);\quad | a g(\cdot, u_0)|_{{1,\Omega_n\cap (u_0\le L)}}\le nL(1+L). $$ \end{lemma} \begin{proof} Denote $X_L=(u_0\le L)$. Multiplying \eqref{e1.3} with $\varepsilon$ replaced by $_{\varepsilon_k}$, by ${{g_{\varepsilon_k}}\over{1+u_{\varepsilon_k}}}$ and integrating on $\Omega_n\cap X_L$: \begin{equation} \label{e3.8} \begin{aligned} &\int_{(\Omega_n\cap X_L)\times\Omega} {{g_{\varepsilon_k}(x)}\over{1+u_{\varepsilon_k}(x)}} K(x,y)g_{\varepsilon_k}(y)dx\,dy\\ & =\int_{\Omega_n\cap X_L}{{g_{\varepsilon_k} u_{\varepsilon_k}}\over{1+u_{\varepsilon_k}}}dx- {\varepsilon_k}^\alpha\int_{\Omega_n\cap X_L} \Delta u_{\varepsilon_k}{{g_{\varepsilon_k}}\over{1+u_{\varepsilon_k}}}dx. \end{aligned} \end{equation} Due to (K2), $$ \int_{(\Omega_n\cap X_L)\times\Omega} {{g_{\varepsilon_k}(x)}\over{1+u_{\varepsilon_k}(x)}} K(x,y)g_{\varepsilon_k}(y)dx\,dy \ge \Big|{{ag_{\varepsilon_k}}\over{1+u_{\varepsilon_k}}}\Big|_{{1,\Omega_n\cap X_L}} \cdot|ag_{\varepsilon_k}|_{{1,\Omega}}; $$ hence from \eqref{e3.8}, \begin{equation} \label{e3.9} \begin{aligned} &\limsup_k\Big(\Big|{{ag_{\varepsilon_k}}\over{1+u_{\varepsilon_k}}} \Big|_{{_{1,\Omega_n\cap X_L}}} \cdot |ag_{\varepsilon_k}|_{{1,\Omega}}\Big)\\ &\le \limsup_k\int_{\Omega_n\cap X_L} {{g_{\varepsilon_k} u_{\varepsilon_k}}\over{1+u_{\varepsilon_k}}}dx+ \limsup_k{\varepsilon_k}^\alpha\int_{\Omega} |\Delta u_{\varepsilon_k}|{{g_{\varepsilon_k}}\over{1+u_{\varepsilon_k}}}dx. \end{aligned} \end{equation} Moreover, by Lemmas \ref{lem3.7} and \ref{lem3.10}, \begin{equation} \label{e3.10} \begin{aligned} \limsup_k\Big(\Big|{{ag_{\varepsilon_k}}\over{1+u_{\varepsilon_k}}} \Big|_{{1,\Omega_n\cap X_L}}\cdot |ag_{\varepsilon_k}|_{{1,\Omega}}\Big) &\ge \limsup_k\Big|{{ag_{\varepsilon_k}}\over{1+u_{\varepsilon_k}}} \Big|_{{1,\Omega_n\cap X_L}}^2\\ &=\Big(\limsup_k \Big|{{ag_{\varepsilon_k}}\over{1+u_{\varepsilon_k}}} \Big|_{{1,\Omega_n\cap X_L}} \Big)^2\\ &= \Big(\limsup_k \Big|{{ag_{\varepsilon_k}}\over{1+u_0}} \Big|_{{1,\Omega_n\cap X_L}}\Big)^2\\ &\ge{1\over{(1+L)^2}} \big(\limsup_k|ag_{\varepsilon_k} |_{{1,\Omega_n\cap X_L}}\big)^2, \end{aligned} \end{equation} \begin{equation} \label{e3.11} \begin{aligned} \limsup_k\int_{\Omega_n\cap X_L} {{g_{\varepsilon_k} u_{\varepsilon_k}}\over{1+u_{\varepsilon_k}}}dx &\le n \limsup_k\int_{\Omega_n\cap X_L} {{ag_{\varepsilon_k} u_{\varepsilon_k}}\over{1+u_{\varepsilon_k}}}dx\\ &=n\limsup_k\int_{\Omega_n\cap X_L} {ag_{\varepsilon_k}} {{u_0}\over{1+u_0}}dx\\ &\le{{nL}\over{1+L}} \limsup_k |ag_{\varepsilon_k}|_{{1,\Omega_n\cap X_L}}. \end{aligned} \end{equation} Finally, since ${\varepsilon_k}\le\bar\varepsilon\le\frac 12 $, we get \begin{align*} {\varepsilon_k}^\alpha\int_\Omega |\Delta u_{\varepsilon_k}|{{g_{\varepsilon_k}}\over{1+u_{\varepsilon_k}}}dx &\le {\varepsilon_k}^\alpha\Big(\int_{({\varepsilon_k} +u_{\varepsilon_k}\le 1)}+ \int_{({\varepsilon_k} +u_{\varepsilon_k}> 1)}\Big)|\Delta u_{\varepsilon_k}| g_{\varepsilon_k} dx\\ &\le{\varepsilon_k}^{\alpha-p}\int_\Omega |\Delta u_{\varepsilon_k}|\phi_0dx+ {\varepsilon_k}^\alpha\int_\Omega |\Delta u_{\varepsilon_k}| g^*(\cdot,1)dx\\ &\le{\varepsilon_k}^{\alpha-p}\sum_{i,j=1}^N \int_\Omega |\partial_{i,j}^2u_{\varepsilon_k} |(\phi_0+g^*(\cdot,1)) dx\\ &\le {\varepsilon_k}^{\alpha- p}\sum_{i,j=1}^N |\partial_{i,j}^2 u_{\varepsilon_k} |_{{q,\Omega}}\cdot |\phi_0+g^*(\cdot,1)|_{{q',\Omega}}, \end{align*} so due to Lemma \ref{lem3.6} \begin{equation} \lim_k {\varepsilon_k}^\alpha\int_\Omega |\Delta u_{\varepsilon_k}| {{g_{\varepsilon_k}}\over{1+u_{\varepsilon_k}}}dx=0.\label{e3.12} \end{equation} Using \eqref{e3.10}, \eqref{e3.11}, \eqref{e3.12} in \eqref{e3.9} $$ {1\over{(1+L)^2}}\big(\limsup_k |ag_{\varepsilon_k}|_{{1,\Omega_n\cap X_L}}\big)^2\le {{nL}\over{1+L}}\limsup_k |ag_{\varepsilon_k}|_{{1,\Omega_n\cap X_L}}, $$ that is the first estimate of the claim. The second one follows from the first one, \eqref{e3.9}and the Fatou's Lemma. \end{proof} \begin{proof}[Proof of Theorem \ref{thm1}] Fix $n, l\in\mathbb{N}^*$, and introduced the notation $Y_l=(u_0\le{1\over{l}})$. From (K2), \begin{align*} \Big|{\eta\over{1+\eta}} (K(g_{\varepsilon_k}\chi_{{\Omega_n\setminus Y_l}}) - v_n)\Big|_{{1,\Omega_n}} &= \Big|{\eta\over{1+\eta}} (K(g_{\varepsilon_k}\chi_{{\Omega_n}})- K(g_{\varepsilon_k}\chi_{{\Omega_n\cap Y_l}}) -v_n)\Big|_{{1,\Omega_n}}\\ &\le |K(g_{\varepsilon_k}\chi_{{\Omega_n}})-v_n|_{{1,\Omega_n}}+ |\eta K(g_{\varepsilon_k}\chi_{{\Omega_n\cap Y_l}})|_{{1,\Omega_n}}\\ &\le |K(g_{\varepsilon_k}\chi_{{\Omega_n}})- v_n|_{{1,\Omega_n}}+ |ag_{\varepsilon_k}|_{{1,\Omega_n\cap Y_l}}. \end{align*} Due to the definition of $v_n$ and Lemma \ref{lem3.11}, sending $k\to +\infty$, we obtain $$ \limsup_k\Big|{\eta\over{1+\eta}} (K(g_{\varepsilon_k}\chi_{{\Omega_n\setminus Y_l}}) -v_n)\Big|_{{1,\Omega_n}}\le {n\over l}(1+{1\over l})\le {{2n}\over l}. $$ Using Fatou's Lemma and \eqref{e3.9}, $$ \Big|{\eta\over{1+\eta}} (K(g(\cdot,u_0)\chi_{{\Omega_n\setminus Y_l}})-v_n)\Big|_{{1,\Omega_n}} \le {{2n}\over l}. $$ Passing to the limit as $l\to +\infty$, $$ v_n(x)=\lim_l\int_{\Omega_n\setminus Y_l}K(x,y)g(y,u_0(y))dy =\int_{\Omega_n\cap(00$. Assume, by contradiction, that $\mathcal{N}=(u_0=0)$ has positive measure. We have that $$ 0=\int_{\Omega\setminus\mathcal{N}}K(x,y)g(y,u_0(y))dy, \quad x\in\mathcal{N}, $$ and using (K2) $$ 0=\int_{\Omega\setminus\mathcal{N}}a(y)g(y,u_0(y))dy. $$ Since $a(y)>0$ a.e. in $\Omega$, we have that $g(y,u_0(y))=0$ in ${\Omega\setminus\mathcal{N}}$. From \eqref{e3.13} $|\mathcal{N}|=|\Omega|$. Due to \eqref{e3.9} we know $u_{\varepsilon_k}\to 0$ a.e. in $\Omega$ and in particular in $\Omega_0$. By fixed $0<\sigma<\mu_0$, in light of (G3) there exists $s_0>0$ such that $$ y\in\Omega_0,\quad 0(\mu_0-\sigma)s. $$ For the reason that $|\Omega_0|<\infty$, there exists $\Omega_\sigma\subset\Omega_0$ such that $|\Omega_\sigma|<\sigma$ and $u_{\varepsilon_k}\to 0$ uniformly in $\Omega_0\setminus\Omega_\sigma$ (Egorov-Severini Theorem). Then $$ k>k_0,\;y\in\Omega_0\setminus\Omega_\sigma\;\Rightarrow\; g_{\varepsilon_k}(y)>(\mu_0-\sigma)({\varepsilon_k} +u_{\varepsilon_k}(y)),\label{e3.14}$$ for some $k_0\in\mathbb{N}$. Multiplying \eqref{e1.3} with $\varepsilon$ replaced by $_{\varepsilon_k}$, by $a_{\lambda}(x):={{a(x)}\over{1+\lambda a(x)}}$ and integrating on $\Omega_0\setminus\Omega_\sigma$ \begin{equation} \label{e3.15} -{\varepsilon_k}^\alpha\int_{\Omega_0\setminus\Omega_\sigma} a_{\lambda}\Delta u_{\varepsilon_k} dx+ \int_{\Omega_0\setminus\Omega_\sigma}a_{\lambda} u_{\varepsilon_k} dx= \int_\Omega g_{\varepsilon_k}(y)dy\int_{\Omega_0\setminus\Omega_\sigma} K(x,y)a_{\lambda}(x) dx. \end{equation} Since $a_{\lambda}\in L^\infty(\Omega)$, Lemma \ref{lem3.6} implies \begin{equation} \label{e3.16} \begin{aligned} -{\varepsilon_k}^\alpha\int_{\Omega_0\setminus\Omega_\sigma} a_{\lambda}\Delta u_{\varepsilon_k} dx &\le{\varepsilon_k}^\alpha \sum_{i,j=1}^N |\partial_{i,j}^2u_{\varepsilon_k}|_{{q,\Omega_0\setminus\Omega_\sigma}} |a_{\lambda}|_{{q',\Omega_0\setminus\Omega_\sigma}} \\ &\le \bar c{\varepsilon_k}^{p+2} |a_{\lambda}|_{{q',\Omega_0\setminus\Omega_\sigma}}\\ &= {{\bar c{\varepsilon_k}^{p+1} |a_{\lambda}|_{{q',\Omega_0\setminus\Omega_\sigma}}} \over{|a_{\lambda}|_{{1,\Omega_0\setminus\Omega_\sigma}}}} \int_{\Omega_0\setminus\Omega_\sigma}a_{\lambda}{\varepsilon_k} dx\\ &\le {{\bar c|a_{\lambda}|_{{q',\Omega}}}\over {|a_{\lambda}|_{{1,\Omega_0\setminus\Omega_\sigma}}}} {\varepsilon_k}^{p+1}\int_{\Omega_0\setminus\Omega_\sigma} a_{\lambda}({\varepsilon_k}+u_{\varepsilon_k})dx. \end{aligned} \end{equation} Using now (K2) and \eqref{e3.14}, for every $k>k_0$, we have that \begin{equation} \int_\Omega g_{\varepsilon_k}(y)dy\int_{\Omega_0\setminus\Omega_\sigma} K(x,y)a_{\lambda}(x)dx\ge (\mu_0-\sigma) \int_{\Omega_0\setminus\Omega_\sigma}a_{\lambda}({\varepsilon_k}+u_{\varepsilon_k})dx \int_{\Omega_0\setminus\Omega_\sigma} aa_{\lambda} dx.\label{e3.17} \end{equation} Substituting \eqref{e3.16}, \eqref{e3.17} in \eqref{e3.15}, \begin{align*} &{{\bar c|a_{\lambda}|_{{q',\Omega}}}\over {|a_{\lambda}|_{{1,\Omega\setminus\Omega_\sigma}}}}{\varepsilon_k}^{p+1} \int_{\Omega_0\setminus\Omega_\sigma} a_{\lambda}({\varepsilon_k}+u_{\varepsilon_k})dx+ \int_{\Omega_0\setminus\Omega_\sigma}a_{\lambda} ({\varepsilon_k}+u_{\varepsilon_k})dx\\ &\ge (\mu_0-\sigma)|aa_{\lambda}|_{{1,\Omega_0\setminus\Omega_\sigma}} \int_{\Omega_0\setminus\Omega_\sigma}a_{\lambda} ({\varepsilon_k}+u_{\varepsilon_k})dx, \end{align*} that gives $$ {{\bar c|a_{\lambda}|_{{q',\Omega}}}\over {|a_{\lambda}|_{{1,\Omega\setminus\Omega_\sigma}}}} {\varepsilon_k}^{p+1}+1\ge(\mu_0-\sigma) |aa_{\lambda}|_{{1,\Omega_0\setminus\Omega_\sigma}}. $$ Sending first $k\to+\infty$, then $\sigma\to 0$ and finally $\lambda\to 0$, we get $$ 1\ge\mu_0 |a^2|_{{1,\Omega_0}}. $$ That contradicts \eqref{e2.1}, hence $u_0>0$ a.e. in $\Omega$. We continue by proving that $u_{\varepsilon_k}\to u_0$ in $L^1_+(\eta,\Omega)$. Reminding that $u_0, u_{\varepsilon_k}$ are solutions of \eqref{e1.1} and \eqref{e1.3} we get \begin{align*} I_k&:=|\eta(u_0-u_{\varepsilon_k})|_{{1,\Omega}}\\ &\le |\eta(K(g(\cdot, u_0))-K(g_{\varepsilon_k}))|_{{1,\Omega}}+ {\varepsilon_k}^\alpha|\eta\Delta u_{\varepsilon_k}|_{{1,\Omega}}\\ &\le |a(g(\cdot, u_0)-g_{\varepsilon_k})|_{{1,\Omega}}+{\varepsilon_k}^\alpha |\eta|_{{q',\Omega}} \sum_{i,j=1}^N |\partial_{i,j}^2u_{\varepsilon_k}|_{{q,\Omega}}. \end{align*} In light of Lemma \ref{lem3.6}, $$ I_k\le |a(g(\cdot,u_0)-g_{\varepsilon_k})|_{{1,\Omega}}+ \bar c|\eta|_{{_{q',\Omega}}}{\varepsilon_k}^{p+2}. $$ Due to the positivity of $u_0$, $ g_{\varepsilon_k}\to g(\cdot,u_0)$ a.e. in $\Omega$; hence the equiabsolute continuity of the integrals in $ag_{\varepsilon_k}$ (see Lemma \ref{lem3.7}) and Vitali's Theorem say \begin{equation} |a(g(\cdot,u_0)-g_{\varepsilon_k})|_{{1,\Omega}}\to 0.\label{e3.18} \end{equation} Therefore, $ I_{{k}}\to 0$, so the claim is proved. We conclude by proving that $|ag(\cdot,u_0)|_{{1,\Omega}}a(x)\le u_0(x)$, $x\in\Omega$ a.e. Due to (K2), \begin{align*} -{\varepsilon_k}^\alpha\Delta u_{{\varepsilon_k}}(x)+ u_{{\varepsilon_k}}(x)\\ &=K(g_{{\varepsilon_k}})(x)\\ &\ge a(x)|ag_{{\varepsilon_k}}|_{{1,\Omega}}\\ &\ge a(x)\big(|ag(\cdot, u_0)|_{{1,\Omega}}- |a\big(g_{{\varepsilon_k}}-g(\cdot, u_0)\big)|_{{1,\Omega}}\big). \end{align*} Since $u_{{\varepsilon_k}}\to u_0$ a.e. in $\Omega$ in light of Lemma \ref{lem3.6} that must be ${\varepsilon_k}^\alpha\Delta u_{{\varepsilon_k}}\to 0$ in $ L^q(\Omega)$, and passing to a subsequence ${\varepsilon_k}^\alpha\Delta u_{{\varepsilon_k}}\to 0$ a.e. in $\Omega$. Hence the claim follows from \eqref{e3.18}. The last part of the statement was proved in Lemma \ref{lem3.6}. \end{proof} \section{Proof of Theorem \ref{thm2}} Let us list some of the properties of the Green's function $G(x,y)$ of the Dirichlet problem for $-\Delta$ in $\Omega$. \begin{lemma}[{\cite[Lemma 3.1]{c1}}] \label{lem4.1} There exists a constant $c_{1}>0$ such that \begin{itemize} \item[(i)] ${{\delta(x)\delta(y)}\over{c_{1}}}\le G(x,y)$; $\int_\Omega G(x,y)dx\le c_{1}\delta(y)$. \item[(ii)] $$ \big(\int_\Omega G(x,y)^\sigma dy\big)^{1\over\sigma} \le c_{1}\int_\Omega G(x,y)dy,\quad 1\le\sigma<{N\over{N-1}}. $$ \item[(iii)] $ |\nabla_x G(x,y)|\le{{c_{1}}\over{| x-y|^{N-1}}}$, $x\neq y$. \end{itemize} \end{lemma} \begin{lemma} \label{lem4.2} Let $\psi\in L^r(\Omega)$, $1N$, then $ G(\psi)\in W^{2,r}(\Omega)\subset C^1(\bar\Omega)$. \end{lemma} We will use a simplified version of the following Agmon's interior regularity result. \begin{lemma}[{\cite[Theorem 7.1]{a1}}] \label{lem4.3} Let $u\in L^\alpha_{\rm loc}(\Omega)$, $1<\alpha$, be such that $\Delta u\in L^\beta_{\rm loc}(\Omega)$, $1<\beta$, where $\Delta u$ is defined by $$ \int_\Omega\Delta u\cdot\phi dx= \int_\Omega u\cdot\Delta\phi dx,\quad \forall\phi\in C^\infty_0(\Omega). $$ Then $u\in W^{2,\beta}_{\rm loc}(\Omega)$ and for every $\Omega'\subset\subset\Omega''\subset\subset\Omega$ there exists $c>0$ such that $$ \Vert u\Vert_{{W^{2,\beta}(\Omega')}}\le c \big(|\Delta u|_{{\beta,\Omega''}}+|u|_{{\beta,\Omega''}}\big). $$ \end{lemma} First of all we prove that the solutions of \eqref{e1.3} are also solutions of \eqref{e2.4}. \begin{lemma} \label{lem4.4} For each $\varepsilon>0$, the solutions to \eqref{e1.3} in $W^{2,q}(\Omega)$ belong to $W^{4,r}(\Omega)$ and solves \eqref{e2.4}. \end{lemma} \begin{proof} Let $u_\varepsilon \in W^{2,q}(\Omega)$ be solution to \eqref{e1.3} with $G(x,y)$ instead of $K(x,y)$ (see Corollary \ref{coro3.2}). Since $g_\varepsilon\in L^r(\Omega)$ (see (G1) and Remark \ref{rmk1}), due to Lemma \ref{lem4.2}, we have that $G(g_\varepsilon)\in W^{2,r}(\Omega)$, hence $u_\varepsilon\in W^{2,r}(\Omega)$. From \eqref{e1.3} $$ \Delta u_\varepsilon=\varepsilon^{-\alpha}(u_\varepsilon-G(g_\varepsilon))\in W^{2,r}(\Omega);\quad \Delta u_\varepsilon=0\text{ on }\partial\Omega. $$ The lemma is established. \end{proof} Due to Theorem \ref{thm1} there exist $u_0\in L^1_+(\Omega)$ and $(\varepsilon_{k})_{{k\in\mathbb{N}}}$, $0<{\varepsilon_k}<\bar\varepsilon\le\frac 12 $, tending to 0, such that \begin{equation} \label{e4.1} \begin{gathered} u_0(x)=\int_\Omega G(x,y) g\big(y,u_0(y)\big)dy,\quad x\in\Omega,\\ \lim_k |u_0- u_{{\varepsilon_k}}|_{{1,\Omega}}=0;\quad u_{{\varepsilon_k}}\to u_0\quad q.o \text{ in }\Omega;\quad u_{{\varepsilon_k}}\in W^{2,q}(\Omega)\cap W^{1,q}_0(\Omega), \end{gathered} \end{equation} and $$ -{\varepsilon_k}^\alpha\Delta u_{{\varepsilon_k}}(x)+u_{{\varepsilon_k}}(x)= \int_\Omega G(x,y)g\big(y,{\varepsilon_k}+u_{{\varepsilon_k}}(y)\big)dy,\quad x\in \Omega. $$ \begin{lemma} \label{lem4.5} There exist $k_0\in\mathbb{N}$ and $c_{2}>0$ such that $$ k_00, $$ there exists $k_0\in\mathbb{N}$ such that $$ k>k_0\>\>\Rightarrow\>\>|\delta g_{\varepsilon_k}|_{{1,\Omega}}>\frac 12 |\delta g(\cdot ,u_0)|_{{1,\Omega}}\>\>\Rightarrow\>\> G(g_{\varepsilon_k})(x) \ge {{\delta(x)}\over{2c_{1}}} |\delta g(\cdot ,u_0)|_{{1,\Omega}}. $$ Let $\phi_{1}$ be a positive eigenfunction and $\lambda_{1}$ be the first eigenvalue of the Dirichlet problem for $-\Delta$ in $\Omega$. Since $\phi_{1}$ and $\delta$ are equivalent in the sense that $$ 0<\inf_{x\in\Omega}{{\delta(x)}\over{\phi_{1}(x)}}< \sup_{x\in\Omega}{{\delta(x)}\over{\phi_{1}(x)}}<\infty, $$ there exists an eigenfunction $\phi_{1}$ relatively to $\lambda_{1}$ such that $$ k>k_0\;\Rightarrow\; G(g_{\varepsilon_k})(x)\ge (\lambda_{1}+1)\phi_{1}(x)> ({\varepsilon_k}^\alpha\lambda_{1}+1)\phi_{1}(x) =-{\varepsilon_k}^\alpha\Delta\phi_{1}(x)+\phi_{1}(x). $$ From \eqref{e1.3} with $\varepsilon$ replaced by ${\varepsilon_k}$, $$ k>k_0\;\Rightarrow\; -{\varepsilon_k}^\alpha\Delta u_{{\varepsilon_k}}+u_{{\varepsilon_k}}\ge -{\varepsilon_k}^\alpha\Delta\phi_{1}+\phi_{1}. $$ Due to the Maximum Principle $u_{{\varepsilon_k}}\ge\phi_{1}, k>k_0$. Finally, since $\phi_{1}$ and $\delta$ are equivalent we get the first estimate of the claim. \end{proof} \begin{lemma} \label{lem4.6} $u_0\in W^{2,r}_{\rm loc}(\Omega)\cap L^q(\Omega)$ and $-\Delta u_0(x)=g(x,u_0(x))$. \end{lemma} \begin{proof} Introduce the notation $v_{\varepsilon_k}:=-\Delta u_{{\varepsilon_k}}$. From Lemma \ref{lem4.4}, \begin{equation} \begin{gathered} -{\varepsilon_k}^\alpha\Delta v_{\varepsilon_k}+v_{\varepsilon_k}=g_{\varepsilon_k}\quad \text{in }\Omega,\\ v_{\varepsilon_k}=0\quad \text{on }\partial\Omega. \end{gathered} \label{e4.2} \end{equation} Hence for a fixed $\phi\in C^2_0(\Omega)$, \begin{equation} -{\varepsilon_k}^\alpha\Delta(\phi v_{\varepsilon_k})+(\phi v_{\varepsilon_k})=f_{\varepsilon_k},\label{e4.3} \end{equation} where $$ f_{\varepsilon_k}:=\phi g_{\varepsilon_k}-{\varepsilon_k}^\alpha(2\nabla\phi\nabla v_{\varepsilon_k}+ v_{\varepsilon_k}\Delta\phi). $$ From Lemma \ref{lem4.5} and (G1), $g_{\varepsilon_k}\in L^r(\Omega)$, $g(\cdot,u_0)\in L^r_{\rm loc}(\Omega)$. We claim that \begin{equation} \lim_{k}|f_{\varepsilon_k}-\phi g(\cdot,u_0)|_{{r,\Omega}}=0.\label{e4.4} \end{equation} Observe that \begin{align*} |f_{\varepsilon_k}-\phi g(\cdot,u_0)|_{{r,\Omega}} &\le |\phi(g_{\varepsilon_k}-g(\cdot,u_0))|_{{r,\Omega}}+{\varepsilon_k}^\alpha |2\nabla \phi\nabla v_{\varepsilon_k}+v_{\varepsilon_k}\Delta \phi|_{{r,\Omega}}\\ &\le |\phi(g_{\varepsilon_k}-g(\cdot,u_0))|_{{r,\Omega}}+\mathop{\rm const}{\varepsilon_k}^\alpha \Vert v_{\varepsilon_k}\Vert_{{W^{1,r}(\Omega)}}. \end{align*} Applying Lemma \ref{lem3.3} to \eqref{e4.2}, \begin{align*} &|f_{\varepsilon_k}-\phi g(\cdot,u_0)|_{{r,\Omega}}\\ &\le |\phi(g_{\varepsilon_k}-g(\cdot,u_0))|_{{r,\Omega}} +\mathop{\rm const}{\varepsilon_k}^{{\alpha\over2}} |g_{\varepsilon_k}|_{{r,\Omega}}\\ &\le |\phi(g_{\varepsilon_k}-g(\cdot,u_0))|_{{r,\Omega}}+ \mathop{\rm const}{\varepsilon_k}^{{\alpha\over2}}\Big(\big(\int_{{\varepsilon_k}+u_{{\varepsilon_k}}\le 1} +\int_{{\varepsilon_k}+u_{{\varepsilon_k}}\ge 1}\big)g_{\varepsilon_k}^rdx\Big)^{1\over r}\\ &\le |\phi(g_{\varepsilon_k}-g(\cdot,u_0))|_{{r,\Omega}}+\mathop{\rm const} \Big({\varepsilon_k}^{r({\alpha\over2}-p)}\int_\Omega\phi_0^rdx+ {\varepsilon_k}^{{{r\alpha}\over2}} \int_\Omega (g^*(x,1))^rdx\Big)^{1\over r}. \end{align*} Recalling that ${\alpha\over 2}-p=5q(p+1)-p>0$ we have \begin{equation} \limsup_{k}|f_{\varepsilon_k}-\phi g(\cdot,u_0)|_{{r,\Omega}}\le \limsup_{k}|\phi(g_{\varepsilon_k}-g(\cdot,u_0))|_{{r,\Omega}}.\label{e4.5} \end{equation} Since $u_{{\varepsilon_k}}\to u_0$ a.e. in $\Omega$ implies $g_{\varepsilon_k}\to g(\cdot,u_0)$ a.e. in $\Omega$ and $\mathop{\rm dist}(\mathop{\rm supp}\phi,\partial\Omega)>0$, from Lemma \ref{lem4.5}, \eqref{e4.1} and the Dominate Convergence Theorem, $$ \lim_k|\phi(g_{\varepsilon_k}-g(\cdot,u_0))|_{{r,\Omega}}=0. $$ Hence \eqref{e4.4} follows from \eqref{e4.5}. Applying Lemma \ref{lem3.3} to \eqref{e4.3}, by \eqref{e4.4}, \begin{equation} \forall \phi\in C^2_0(\Omega):\quad \lim_k |\phi(v_{{\varepsilon_k}}-g(\cdot,u_0))|_{{r,\Omega}}=0.\label{e4.6} \end{equation} Observing that \eqref{e4.1}, \eqref{e4.6} give \begin{align*} \int_\Omega\phi(x) g(x, u_0)dx &= \lim_k \int_\Omega \phi v_{\varepsilon_k} dx \\ &= \lim_k \int_\Omega\phi (-\Delta u_{{\varepsilon_k}}) dx \\ &= \lim_k \int_\Omega (-\Delta \phi)u_{{\varepsilon_k}} dx \\ &= \int_\Omega(-\Delta\phi) u_0 dx; \end{align*} therefore, \begin{equation} -\Delta u_0=g(\cdot,u_0)\quad \text{in }\Omega \text{ in the sense of distributions.} \label{e4.7} \end{equation} We claim that \begin{equation} u_0\in L^q(\Omega).\label{e4.8} \end{equation} Observe that \begin{align*} u_0(x)&=\int_\Omega G(x,y)g(y,u_0(y))dy\\ &= \int_\Omega{{G(x,y)}\over{\delta(y)^{1\over{q'}}}} g(y,u_0(y))^{1\over q}(\delta (y)g(y,u_0(y)))^{1\over {q'}}dy\\ &\le \Big(\int_\Omega{{G(x,y)^q}\over{\delta(y)^{q-1}}} g(y,u_0(y))dy\Big)^{1\over q}|\delta g(\cdot,u_0)|^{1\over {q'}}_{{1,\Omega}}. \end{align*} Hence \begin{align*} |u_0|_{{q,\Omega}}^q &\le |\delta g(\cdot,u_0)|^{q-1}_{{1,\Omega}} \int_\Omega dx\int_\Omega{{G(x,y)^q}\over{\delta(y)^{q-1}}} g(y,u_0(y))dy\\ &=|\delta g(\cdot,u_0)|^{q-1}_{{1,\Omega}} \int_\Omega{{\delta(y)g(y,u_0(y))}\over{\delta^q(y)}}dy \int_\Omega G(x,y)^qdx. \end{align*} From \eqref{e4.1}(i) and \eqref{e4.1}(ii), $$ |u_0|_{{q,\Omega}}^q\le |\delta g(\cdot,u_0)|^{q-1}_{{1,\Omega}} c_{1}^{2q} \int_\Omega{{\delta(y)g(y,u_0(y))}\over{\delta^q (y)}} \delta^q(y)dy=c_{1}^{2q}|\delta g(\cdot,u_0)|^{q}_{{1,\Omega}}. $$ In light of Theorem \ref{thm1}, $g(\cdot, u_0)\in L^1_+(\delta,\Omega)$, hence \eqref{e4.8} is true. We need to prove that $$ u_0\in W^{2,r}_{\rm loc}(\Omega). $$ Since \eqref{e4.7} holds in the sense of distributions we have simply to apply Lemma \ref{lem4.3} with $\alpha=q$ and $\beta =r$. The lemma is proved. \end{proof} \begin{lemma} \label{lem4.7} $u_0\in C(\bar\Omega)$ and $ u_0(x)=0, x\in\partial \Omega$. \end{lemma} \begin{proof} Since $q<{{N}\over{N-1}}$ and $q+r\le qr$ give $r>N$, due to Lemmas \ref{lem4.5} and \ref{lem4.6}, $0\le c_{2}\delta\le u_0\in C(\Omega)$. Therefore, we have only to prove that \begin{equation} \lim_{x\to x_0}u_0(x)=0,\quad x_0\in\partial\Omega.\label{e4.9} \end{equation} Define $\theta_{1}:\mathbb{R}_+\to \mathbb{R}$ by $$ \theta_{1}(t)= \begin{cases} ([p]+2)([p]+1)t^{[p]}& \text{if }0\le t\le 3,\\ 0& \text{if } t> 3, \end{cases} $$ where $[p]$ is the integer part of $p$, and $\theta_{2}\in C^\infty(\mathbb{R}_+)$ such that $$ 0\le \theta_{2}\le 1,\quad 0\le t\le 1\Rightarrow\theta_{2}(t)=1,\quad t\ge 2\Rightarrow\theta_{2}(t)=0. $$ Denoting $$ \theta(t):=\int_0^tdt_{1}\int_0^{t_{1}} \theta_{1}(\tau)\theta_{2}(\tau)d\tau,\quad t\in\mathbb{R}_+, $$ observe that \begin{gather*} \theta\ge 0;\quad \theta(t)=t^{[p]+2},\quad 0\le t\le 1;\\ \theta'(t)=\int_0^t\theta_{1}(\tau)\theta_{2}(\tau)d\tau\le \theta'(2);\quad \theta''=\theta_{1}\theta_{2}\ge0;\quad \theta\in C^\infty(\mathbb{R}_+). \end{gather*} Since $\theta''\ge 0$, $-\Delta (\theta(u_{\varepsilon_k}))\le \theta'(u_{\varepsilon_k})(-\Delta u_{\varepsilon_k})$. As in the Proof of Lemma \ref{lem4.6}, denoting $-\Delta u_{\varepsilon_k}=v_{\varepsilon_k}$, from \eqref{e2.4} with ${\varepsilon_k}$ instead of $\varepsilon$, we get $$ -\Delta u_{\varepsilon_k}=g_{\varepsilon_k}-{\varepsilon_k}^\alpha\Delta^2 u_{\varepsilon_k}= g_{\varepsilon_k}+{\varepsilon_k}^\alpha\Delta v_{\varepsilon_k}. $$ Therefore, \begin{equation} -\Delta(\theta(u_{\varepsilon_k}))\le \theta'(u_{\varepsilon_k})(g_{\varepsilon_k}+{\varepsilon_k}^\alpha \Delta v_{\varepsilon_k}).\label{e4.10} \end{equation} Since $r>N$, $u_{\varepsilon_k}\in W^{4,r}(\Omega)\subset C^3(\bar\Omega)$ and $u_{\varepsilon_k}=0$ on $\partial \Omega$, from the properties of $\theta$ we deduce that $\theta(u_{\varepsilon_k})\in W^{4,r}(\Omega)$ and $\theta(u_{\varepsilon_k})=0$ on $\partial \Omega$. Thanks to the positivity of the Green's function from \eqref{e4.10}, we get \begin{equation} \theta(u_{\varepsilon_k})\le G(\theta'(u_{\varepsilon_k})g_{\varepsilon_k})+{\varepsilon_k}^\alpha G(\theta'(u_{\varepsilon_k})\Delta v_{\varepsilon_k}).\label{e4.11} \end{equation} Integrating by parts \begin{align*} G(\theta'(u_{\varepsilon_k})\Delta v_{\varepsilon_k})(x) &= -\int_\Omega\nabla_yG(x,y) \theta'(u_{\varepsilon_k}(y))\nabla v_{\varepsilon_k}(y)dy\\ &\quad -\int_\Omega G(x,y) \nabla(\theta'(u_{\varepsilon_k}(y)))\nabla v_{\varepsilon_k}(y)dy, \end{align*} so \begin{align*} I_k&:={\varepsilon_k}^\alpha|G(\theta'(u_{\varepsilon_k})\Delta v_{\varepsilon_k})|_{{1,\Omega}}\\ & \le \int_{\Omega\times\Omega}|\nabla_yG(x,y)| \theta'(u_{\varepsilon_k}(y))|\nabla v_{\varepsilon_k}(y)|dx\,dy \\ &\quad + \int_{\Omega\times\Omega} G(x,y) |\nabla(\theta'(u_{\varepsilon_k}(y)))|\cdot|\nabla v_{\varepsilon_k}(y)|dx\,dy. \end{align*} Since $$ \sup_{y\in\Omega}\Big(\int_\Omega|\nabla_y G(x,y)|dx+ \int_\Omega G(x,y)dx\Big)<\infty, $$ the boundedness of $\theta'$ and $\theta''$ implies $$ I_k\le \mathop{\rm const}{\varepsilon_k}^\alpha\int_\Omega (1+|\nabla u_{\varepsilon_k}|)|\nabla v_{\varepsilon_k}|dy\le \mathop{\rm const}{\varepsilon_k}^\alpha(|\nabla v_{\varepsilon_k}|_{{1,\Omega}}+ |\nabla u_{\varepsilon_k}|_{{q,\Omega}}|\nabla v_{\varepsilon_k}|_{{q',\Omega}}). $$ Since $q'\le r$, $$ I_k\le \mathop{\rm const}{\varepsilon_k}^\alpha(1+\sum_{i=1}^N|\partial_i u_{\varepsilon_k}|_{{q,\Omega}}) \Vert v_{\varepsilon_k}\Vert_{{W^{1,r}(\Omega)}}, $$ applying Theorem \ref{thm1} and Lemma \ref{lem3.3} to \eqref{e2.4} with ${\varepsilon_k}$ instead of $\varepsilon$, $$ I_k\le \mathop{\rm const}({\varepsilon_k}^{\alpha\over 2}+{\varepsilon_k}^{p+2})|g_{\varepsilon_k}|_{{r,\Omega}} \le \mathop{\rm const}({\varepsilon_k}^{\alpha\over 2}+{\varepsilon_k}^{p+2}) \Big({{|\phi_0|_{{r,\Omega}}}\over{{\varepsilon_k}^p}}+|g^*(\cdot,1)|_{{r,\Omega}}\Big). $$ Hence \begin{equation} \lim_k I_k=0.\label{e4.12} \end{equation} Observe that $(\theta'(u_{\varepsilon_k})g_{\varepsilon_k})_{{k\in\mathbb{N}}}$ is bounded in $L^r(\Omega)$. Indeed from (G1) and the properties of $\theta'$, \begin{align*} |\theta'(u_{\varepsilon_k})g_{\varepsilon_k}|_{{r,\Omega}} &\le |\theta'(u_{\varepsilon_k})g_{\varepsilon_k}\chi_{{({\varepsilon_k}+u_{\varepsilon_k}\le 1)}}|_{{r,\Omega}}+ |\theta'(u_{\varepsilon_k})g_{\varepsilon_k}\chi_{{({\varepsilon_k}+u_{\varepsilon_k}\ge 1)}}|_{{r,\Omega}}\\ &\le\Big|{{([p]+2)u_{\varepsilon_k}^{[p]+1}\phi_0 \chi_{{({\varepsilon_k}+u_{\varepsilon_k}\le 1)}}} \over{({\varepsilon_k}+u_{\varepsilon_k})^p}}\Big|_{{r,\Omega}}+ |\theta'(2)g^*(\cdot,1)|_{{r,\Omega}}\\ &\le \mathop{\rm const}(|\phi_0|_{{r,\Omega}}+|g^*(\cdot,1)|_{{r,\Omega}}). \end{align*} Fatou's Lemma and Vitali's Theorem give \begin{equation} \theta'(u_0)g(\cdot,u_0)\in L^r(\Omega);\quad \lim_k G(\theta'(u_{\varepsilon_k})g_{\varepsilon_k})=G(\theta'(u_0)g(\cdot,u_0)). \label{e4.13} \end{equation} Hence, from \eqref{e4.11} and \eqref{e4.12} we get $$ 0\le \theta(u_0)\le G(\theta'(u_0)g(\cdot,u_0)). $$ Since $r>N$, by \eqref{e4.13} $G(\theta'(u_0)g(\cdot,u_0))\in C^1(\bar\Omega)$ and $G(\theta'(u_0)g(\cdot,u_0))=0$ on $\partial\Omega$, it follows that $$ \lim_{x\to x_0}\theta(u_0(x))=0,\quad x_0\in\partial\Omega. $$ Then the monotonicity of $\theta$ implies \eqref{e4.9}. \end{proof} \begin{proof}[Proof of Theorem \ref{thm2}] In light of previous lemmas we have to prove that $u_{{\varepsilon_k}}\to u_0$ in $ W^{2,r}_{\rm loc}(\Omega)\cap L^q(\Omega)$. We begin by proving \begin{equation} u_{{\varepsilon_k}}\to u_0\>\>{\rm in}\>\> L^q(\Omega).\label{e4.14} \end{equation} Observing that $$ -{\varepsilon_k}^\alpha\Delta u_{{\varepsilon_k}}+(u_{{\varepsilon_k}}-u_0)=G(g_{{\varepsilon_k}}-g(\cdot,u_0)), $$ and using the same argument of the proof of \eqref{e4.8}, from Lemma \ref{lem3.6}, $$ |u_{{\varepsilon_k}}-u_0|_{{q,\Omega}}\le |G(g_{{\varepsilon_k}}-g(\cdot,u_0))|_{{q,\Omega}}+{\varepsilon_k}^\alpha |\Delta u_{{\varepsilon_k}} |_{{q,\Omega}}\le c_{1}^2 |\delta(g_{{\varepsilon_k}}-g(\cdot,u_0))|_{{1,\Omega}}+\bar c{\varepsilon_k}^{p+2}. $$ Since the integrals that define $\delta g_{{\varepsilon_k}}$ are equiabsolutely continuous (see Lemma \ref{lem3.7}) and $g_{{\varepsilon_k}}\to g(\cdot,u_0)$ a.e. in $\Omega$, Vitali's Theorem gives \eqref{e4.14}. We continue by proving that \begin{equation} u_{{\varepsilon_k}}\to u_0\>\>{\rm in}\>\> W^{2,r}_{\rm loc}(\Omega).\label{e4.15} \end{equation} Let $\Omega',\Omega''$ be two open subsets of $\Omega$ such that $\Omega'\subset\subset\Omega''\subset\subset\Omega$. Denoting $v_{{\varepsilon_k}}=-\Delta u_{{\varepsilon_k}}$ from \eqref{e2.3}, \eqref{e2.4}, we get \begin{equation} -\Delta(u_{{\varepsilon_k}}-u_0)=g_{{\varepsilon_k}}-g(\cdot,u_0)+ {\varepsilon_k}^\alpha\Delta v_{{\varepsilon_k}}.\label{e4.16} \end{equation} Introducing the notation $$ J_{k}(\Omega^*)=|g_{{\varepsilon_k}}-g(\cdot,u_0)|_{{r,\Omega^*}}+ {\varepsilon_k}^\alpha|\Delta v_{{\varepsilon_k}}|_{{r,\Omega^*}};\quad \delta_k=u_{{\varepsilon_k}}-u_0;\quad \Omega^*\subset\subset\Omega, $$ applying Lemma \ref{lem4.3} with $\alpha=\beta=q$, \begin{equation} \label{e4.17} \begin{aligned} \Vert u_{{\varepsilon_k}}-u_0\Vert_{{W^{2,q}(\Omega')}} &\leq\mathop{\rm const}\big( |g_{{\varepsilon_k}}-g(\cdot,u_0)|_{{q,\Omega''}}+ {\varepsilon_k}^\alpha |\Delta v_{{\varepsilon_k}}|_{{q,\Omega''}} +|u_{{\varepsilon_k}}-u_0|_{{q,\Omega''}}\big)\\ &\leq \mathop{\rm const}(J_k(\Omega'')+|\delta_k|_{{q,\Omega}}). \end{aligned} \end{equation} If $N=2$ we have ${N\over 2}=1q_{l}$, $$ \Vert\delta_k\Vert_{{W^{2,q_{l}}(\Omega')}}\le\mathop{\rm const} (J_k(\Omega'')+|\delta_k|_{{q,\Omega}}), $$ which gives \eqref{e4.18}. In conclusion both \eqref{e4.18} and \eqref{e4.20} imply \eqref{e4.21}. Applying Lemma \ref{lem4.3} to \eqref{e4.16} with $\alpha=q$ and $\beta=r$, \begin{equation} \Vert\delta_k\Vert_{{W^{2,r}(\Omega')}}\le\mathop{\rm const} (J_k(\Omega'')+|\delta_k|_{{q,\Omega}}).\label{e4.22} \end{equation} Since $g_{{\varepsilon_k}}\to g(\cdot,u_0)$ a.e. in $\Omega$, from Lemma \ref{lem4.5} and (G1), $$ \lim_k |g_{{\varepsilon_k}}-g(\cdot,u_0)|_{{r,\Omega''}}=0. $$ Thanks to \eqref{e4.4}, applying Lemma \ref{lem3.3} to \eqref{e4.3} for each $\phi\in C^\infty_0(\mathbb{R}^N)$, with $\phi(x)=1$ for $x\in\Omega''$, we have $$ \lim_k{\varepsilon_k}^\alpha |\Delta u_{{\varepsilon_k}}|_{{r,\Omega''}}\le \lim_k{\varepsilon_k}^\alpha |\Delta(\phi u_{{\varepsilon_k}})|_{{r,\Omega}}= 0. $$ Hence $\lim_k J_k(\Omega'')=0$. Finally, in light of \eqref{e4.14}, \eqref{e4.15} follows from \eqref{e4.22}. \end{proof} \section{Appendix} In this appendix for the sake of completeness, we prove the following result from which Lemma \ref{lem3.10} follows. \begin{lemma} \label{lemA.1} Let $f_{k}\in L^p(\Omega)$, $\phi_{k}\in L^q(\Omega)$, $k\in\mathbb{N}$, $1\le p< \infty$, $1< q\le\infty$, and $|\Omega|<\infty$. If $(f_{k})_{{k\in\mathbb{N}}}$ is bounded in $L^p(\Omega)$, $(\phi_{k})_{{k\in\mathbb{N}}}$ is bounded in $L^q(\Omega)$, is converging in measure to $\phi\in L^q(\Omega)$ and \begin{equation} q'0$, it follows that \begin{equation} \lim_k |\Omega_{\sigma, k}|=0\label{eA3} \end{equation} and \begin{equation} |f_{k}(\phi_{k}-\phi)|_{{1,\Omega}}= |f_{k}(\phi_{k}-\phi)|_{{1,\Omega_{\sigma,k}}} +|f_{k}(\phi_{k}-\phi)|_{{1,\Omega\setminus\Omega_{\sigma,k}}},\label{eA4} \end{equation} where $\Omega_{\sigma, k}:=\big\{x\in\Omega: |\phi_{k}(x)-\phi(x)|>\sigma\big\}$. We begin by considering the case in which \eqref{eA1} holds. Since $10$ such that $$ |E|<\delta\>\Rightarrow\>\int_E|f_{k}|^pdx<\sigma^p,\quad k\in\mathbb{N}. $$ Moreover, due to \eqref{eA3}, there is $k_0\in\mathbb{N}$ such that $|\Omega_{\sigma,k}|<\delta$, $k>k_0$. Therefore, $$ k_0