\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2010(2010), No. 12, pp. 1--12.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2010 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2010/12\hfil Existence of solutions] {Existence of solutions to singular elliptic equations with convection terms via the Galerkin method} \author[C. O. Alves, P. C. Carri\~ao, L. F. O. Faria\hfil EJDE-2010/12\hfilneg] {Claudianor O. Alves, Paulo C. Carri\~ao, Luiz F. O. Faria} % in alphabetical order \address{Claudianor O. Alves \newline Unidade Acad\^emica de Matem\'atica e Estat\'istica\\ Universidade Federal de Campina Grande \\ 58109-970, Campina Grande - PB, Brazil} \email{coalves@dme.ufcg.edu.br} \address{Paulo C. Carri\~ao \newline Departamento de Matem\'atica - Instituto de Ci\^{e}ncias Exatas \\ Universidade Federal de Minas Gerais \\ 30161-970, Belo Horizonte - MG, Brazil} \email{carrion@mat.ufmg.br} \address{Luiz F. O. Faria \newline Departamento de Matem\'atica - Instituto de Ci\^{e}ncias Exatas \\ Universidade Federal de Juiz de Fora \\ 30161-970, Juiz de Fora - MG, Brazil} \email{luiz.faria@ufjf.edu.br} \thanks{Submitted June 22, 2009. Published June 18, 2010.} \thanks{C. O. Alves is supported by grants 472281/2006-2 from FAPESP,\hfill\break\indent and 620025/2006-9 from CNPq. L.F.O. Faria is supported a grant from by FAPEMIG.} \subjclass[2000]{35J60, 35B25} \keywords{Singular elliptic equation; convection term; Galerkin method} \begin{abstract} In this article, we use the Galerkin method to show the existence of solutions for the following elliptic equation with convection term $$ - \Delta u= h(x,u)+\lambda g(x,\nabla u) \quad u(x)>0 \quad \text{in } \Omega, \quad u=0 \quad \text{on } \partial \Omega, $$ where $\Omega$ is a bounded domain, $\lambda \geq 0$ is a parameter, $h$ has sublinear and singular terms, and $g$ is a continuous function. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \newtheorem{example}[theorem]{Example} \newtheorem{definition}[theorem]{Definition} \newtheorem{claim}[theorem]{Claim} \section{Introduction} In this article, we study the existence of solution the problem \begin{equation} \begin{gathered} - \Delta u= h(x,u)+\lambda g(x,\nabla u) \quad \text{in } \Omega,\\ u(x)>0 \quad \text{in } \Omega \\ u=0 \quad \text{on } \partial \Omega \end{gathered}\label{eP} \end{equation} where $\Omega$ is a bounded smooth domain in $\mathbb{R}^{N}$, $N\geq 1$, $\lambda$ is a positive parameter, the function $h$ has sublinear and singular terms and $g$ is a continuous function. By a solution to \eqref{eP}, we mean a $u$ function if $u \in C^{2}(\Omega) \cap H_0^1(\Omega)$ which is positive and satisfies the equation in the classical sense in $\Omega$. Nonlinear singular boundary value problem arise in several physical situations such as fluid mechanics, pseudoplastics flow, chemical heterogeneous catalysts, non-Newtonian fluids, biological pattern formation, for more details about this subject, we cite the papers of Fulks and Maybe \cite{FM}, Callegari and Nashman \cite{CN1,CN2} and the references therein. When the nonlinearity has not a convection term, that is $g = 0$, there exist a lot of work related to this subject; see for example, Crandall, Rabinowitz and Tartar \cite{CRT}, D\'avila and Montenegro \cite{DM}, Choi and McKenna \cite{CM2}, Coclite and Palmieri \cite{CP}, C\^irstea, Ghergu and Radulescu \cite{CGR}, Diaz, Morel and Oswald \cite{DMO}, Alves and Corr\^ea \cite{AC}, Alves, Corr\^ea and Gon\c calves \cite{ACG}. The main tools used in the above papers are Sub and Supersolution, Fixed Point Theorems, Bifurcation Theory and Galerkin Method. When the nonlinearity has a convection term, that is $g \not=0$, we would like to cite the papers of Ghergu and Radulescu \cite{GR1,GR2}, Zhang \cite{Zhang}, Giarrusso and Porru \cite{GP}, Wood \cite{Wood} and references therein. In all these papers, the main tools used are again Sub and Supersolution and Fixed Point Theorem. On the other hand, using variational technique, recently, de Figueiredo, Girardi and Matzeu \cite{FGM} studied elliptic problems, where the nonlinearity depends of the gradient of the solution. In general the above papers assume that $h$ is monotone and $g$ is homogeneous. Here, we show that the Galerkin method can be used to find solutions to \eqref{eP} for nonlinearities where $h$ is not monotone and $g$ is not homogeneous. This way, we believe that our main results can be see as a complement of the study made in the above papers. Moreover, the method used in the present paper can be used to study singular elliptic equations in $\mathbb{R}^{N}$ and elliptic systems with convection terms, see examples 3 and 4 below. The basic hypotheses on functions $h$ and $g$ are the following: \begin{itemize} \item[(H1)] The functions $h:\Omega\times \mathbb{R} \to \mathbb{R}$ and $g:\Omega\times \mathbb{R}^{N}\to \mathbb{R}$ are locally H\"older continuous. \item[(H2)] There exist constants $b>0$, $0 0$, we consider the problem \begin{equation} \label{ePe} \begin{gathered} -\Delta u=h(x,|u|+\epsilon)+\lambda g(x,\nabla u) \quad\text{in } \Omega\\ u>0\quad \text{in } \Omega \\ u=0 \quad \text{on } \partial \Omega \end{gathered} \end{equation} Now, problem \eqref{ePe} can be treated by Galerkin method. We will show that taking $\epsilon=1/n $, it is possible to obtain a family of bounded solutions $\{u_{n}\}$ in $H_0^1(\Omega)$. By passing to the limit in \eqref{ePe} with $\epsilon=1/n$. As $n\to\infty$, we will obtain a solution of \eqref{eP}, strictly positive by a result due to Ambrosetti, Br\'ezis and Cerami \cite{ambbrece}. The first result of this paper is the following. \begin{theorem}\label{TP} If {\rm (H1)-(H3)} hold, then \eqref{eP} has a solution for all $\lambda \geq 0$. \end{theorem} The second result is related with the following hypotheses on $g$: \begin{itemize} \item[(H4)] The conditions of (H2) hold with $r_2 < \min\{ \frac{4}{N-2},1\}$ when $N \geq 3$ and $$ \frac{a_{3}}{\phi_1^{r_{3}}} \in L^{p}(\Omega) \quad \text{for some } p > \frac{N}{2} $$ where $\phi_1$ is a positive eigenfunction corresponding to the first eigenvalue of $(-\Delta, H_0^1(\Omega))$. \item[(H5)] There exists a local H\"older continuous function $g:[0, +\infty) \to [0, +\infty)$ such that $$ g(x,\eta)=g(|\eta|) \,\,\, \forall \eta \in \mathbb{R}^{N} $$ with $g(0)=0$ and $g(t) >0$ for all $t>0$. \item[(H6)] There exists $M^{*}>0$ and $\beta \in (0, \frac{2}{N})$ such that $$ \sup_{t \in (0,M)}\frac{g(t)}{t^{\beta}}=\frac{g(M)}{M^{\beta}}, \quad \forall M \geq M^{*}. $$ \end{itemize} \begin{theorem} \label{T2} Assume that {\rm (H1),(H4), (H5), (H6)} hold. Then there exists $\lambda^{*}>0$ such that \eqref{eP} has a solution for all $0 \leq \lambda \leq \lambda^{*}$. \end{theorem} In the sequel, we show some class of problems where our main theorems can be applied to get a positive solution. These problems were not considered in the above references. \begin{example} \label{exa1}\rm Theorem \ref{TP} establishes the existence of solutions for the problem \begin{gather*} -\Delta u= h(u) + \lambda g(|\nabla u|) \quad\text{in } \Omega\\ u>0\quad \text{in } \Omega, \quad u=0 \quad \text{on } \partial \end{gather*} for all $\lambda >0$, where the functions $g,f$ and $h$ are of the type \begin{gather*} g(t)=\begin{cases} t^{\beta_1}, & 0 \leq t \leq 2 \\ 2^{\beta_1}+\frac{(5^{\beta_2}-2^{\beta_1})}{5-2^{\beta_1}} (t-2^{\beta_1}), & 2 \leq t \leq 5 \\ t^{\beta_2}, & t \geq 5 \end{cases} \\ h(t)=t^{\alpha_1} + \sum_{i=2}^{m}(\cos(it)+1)t^{\alpha_{i}} + \sum_{i=1}^{j}t^{-\gamma_{i}} \end{gather*} where $0< \beta_1< \beta_2<1$, $j, m \in \mathbb{N}$ and $\alpha_{i}, \gamma_{i} \in (0,1)$. \end{example} \begin{example} \label{exa2} \rm Theorem \ref{T2} can be used to prove the existence of solution for the problem \begin{gather*} -\Delta u= h(u) +\lambda g(|\nabla u|) \quad\text{in } \Omega\\ u>0\quad\text{in } \Omega, \quad u=0 \quad \text{on } \partial \Omega \end{gather*} has a solution when $\lambda>0$ is small enough, if $g$ and $h$ are of the form $$ g(t) = \sum_{i=1}^{j}t^{\beta_{i}} \quad h(t)=t^{\alpha_1} + \sum_{k=2}^{m}(\cos(it)+1)t^{\alpha_{i}}, $$ for some $j,m \in \mathbb{N}$, $0<\beta_{i}$, for $i=1,\dots,j$, and $\alpha_{k} \in (0,1)$ for $k=1,\dots,m$. \end{example} \begin{remark} \label{rmk1.1} \rm Note that we can not apply the Theorem \ref{TP} to show the existence of solution for the Example 2, because in this example we do not assume that $\beta_{i} \in (0,1)$. \end{remark} The method used in this paper can be applied to prove the existence of solutions for some class of elliptic systems or elliptic problems in whole $\mathbb{R}^{N}$; for example, Theorem \ref{TP} can be used to establish the existence of solutions in the following two examples. \begin{example} \label{exa3} \rm \begin{gather*} -\Delta u= \frac{1}{u^{\alpha}}+h_1(x,u,v) +g_1(x,\nabla u, \nabla v) \quad \text{in } \Omega \\ - \Delta v= \frac{1}{v^{\beta}}+h_2(x,u,v)+g_2(x,\nabla u, \nabla v) \quad \text{in } \Omega \\ u(x),v(x)>0 \quad \forall x \in \Omega \\ u=v=0 \quad \text{on } \partial \Omega \end{gather*} where $h_1$ and $h_2$ are positive functions, which have a subcritical growth at the variables $u$ and $v$, that is, they are bounded from above by $|u|^{\alpha_1}| v|^{\alpha_2}$ with $\alpha_1,\alpha_2>0$ and $ \alpha_1+\alpha_2<1$. Related to functions $g_1$ and $g_2$, we can assume that they are positive and bounded from above by $|\nabla u|^{\beta_1}|\nabla v|^{\beta_2}$ with $\beta_1,\beta_2>0$ and $ \beta_1+\beta_2<1$. \end{example} \begin{example} \label{exa4} \rm $$ -\Delta{u}=p(x)(h(x,u)+ g(x,\nabla u)), \quad \mathbb{R}^{N} $$ where $p$ is a positive weight satisfying some suitable conditions growth at infinite, while that functions $h$ and $g$ satisfy assumptions of the type (H1)--(H3). Moreover, in this class of problems we use the Sobolev space $D^{1,2}(\mathbb{R}^{N})$. \end{example} To conclude this introduction, we would like to mention that the class of problems cited in examples 3 and 4 complete the study made in the papers \cite{AC}, \cite{ACG} and \cite{GR2}, in the following sense : In \cite{AC} and \cite{ACG} the nonlinearities have not convection term, and in \cite{GR2} the convection term is homogeneous, while in the present paper this hypothesis is not assumed. \section{Preliminary Results} In this section, we will present some results already known and that will be used in the next section. The lemma below is a consequence of Brouwer's Fixed Point Theorem and its proof can be found in Kesavan \cite{kesavan}. \begin{lemma}\label{le1} Let $F:\mathbb{R}^{K}\to\mathbb{R}^{K}$ a continuous function with $\langle F(x),x\rangle\geq0$, for $x$ satisfying $|x|=R>0$, where $\langle x,y\rangle$ is the usual inner product of $\mathbb{R}^{K}$. So, there exists $z_0\in \overline{B}_{R}(0)$ such that $F(z_0)=0$. \end{lemma} Next, we state a result of sub and supersolution due to Ambrosetti, Br\'ezis and Cerami \cite{ambbrece}. Consider the following problem \begin{equation} \begin{gathered} -\Delta v=f(v),\quad \text{in } \Omega\\ v>0,\quad \text{in } \Omega\\ v(x)=0, \quad \text{on } \partial \Omega \end{gathered} \label{eA} \end{equation} We say that $v_1\in C^{2}(\Omega) \cap C(\overline{\Omega})$ is a subsolution of \eqref{eA} if \begin{equation} \begin{gathered} -\Delta v_1\leq f(v_1) \quad \text{in } \Omega\\ v_1>0 \quad \text{in } \Omega\\ v_1(x)=0 \quad \text{on } \partial \Omega\,. \end{gathered} \label{eA1} \end{equation} Similarly, $v_2\in C^{2}(\Omega) \cap C(\overline{\Omega})$ is a supersolution of \eqref{eA} if \begin{equation} \begin{gathered} -\Delta v_2\geq f(v_2) \quad \text{in } \Omega\\ v_2>0 \quad \text{in } \Omega\\ v_2(x)=0 \quad \text{on } \partial \Omega\,. \end{gathered} \label{eA2} \end{equation} \begin{theorem}\label{teorsubsup} Let $f:\mathbb{R}\to\mathbb{R}$ such that $t^{-1}f(t)$ is decreasing for $t>0$. Let $v_1$ and $v_2$ satisfying \eqref{eA1} and \eqref{eA2}, respectively. So, $v_2\geq v_1$ in $\Omega$. \end{theorem} \section{Existence of solutions to \eqref{ePe}} The first lemma of this section is related with the regularity of the weak solutions to \eqref{ePe}. \begin{lemma}[Regularity] \label{leregularidade} Assume that {\rm (H1)--(H3)} hold and let $u\in H_0^1(\Omega)$ be a weak solution of \eqref{ePe}. Then $u$ belongs to $C^{2}(\Omega) \cap C^1(\overline{\Omega})$. \end{lemma} \begin{proof} Define $\Phi(x)=h(x,|u| + \epsilon)+\lambda g(x,\nabla u)$. Since $u\in H_0^1(\Omega)$, by (H2)--(H3), $\Phi\in L^{2/r}(\Omega)$, where $r=\max\{r_{i}, i=2,4\}$. Thus, by Agmon \cite[Theorem 8.2]{Ag}, all solution of \begin{gather*} -\Delta u(x)=\Phi(x) \quad \text{in } \Omega\\ u(x)=0 \quad \text{on } \partial \Omega \end{gather*} belong to $W^{2,s_1}(\Omega)$, where $s_1=2/r$ and therefore $u,\nabla u\in L^{s_1}(\Omega)$ and $\Phi\in L^{s_1/r}(\Omega)$. Using again \cite{Ag}, we obtain $u\in W^{2,s_2}(\Omega)$ with $s_2=2/r^{2}$. Since $r \in (0,1)$, repeating this argument k times, such that $s_{k}=2/r^{k} > N/2$, it follows from the Sobolev embedding that $u$ in $C^{1,\alpha}(\overline{\Omega})$, for some $0<\alpha<1$. Thus, by Schauder regularity theorem and (H1), we conclude that $u\in C^{2}(\Omega)$. \end{proof} To obtain a solution to \eqref{ePe}, we will apply Galerkin method. Note that, under assumptions (H2)--(H3) and the Maximum Principle, any classical solution $u_{\epsilon}$ of \eqref{ePe} is positive; that is, $u_{\epsilon}(x)>0$ for all $x\in\Omega$. \begin{theorem}\label{teo1} Assume {\rm (H1)--(H3)}. Then \eqref{ePe} has a solution $u_{\epsilon}$ in $C^{2}(\Omega) \cap C^1(\overline{\Omega})$. \end{theorem} \begin{proof} Let $\Sigma=\{e_1,\dots,e_{m},\dots\}$ be a orthonormal bases of the Hilbert space $H_0^1(\Omega)$. For each $m\in \mathbb{N}$, define the subspace $V_{m}=[e_1,\dots,e_{m}]$; that is, $V_{m}$ is a $m-$dimensional space generated by the orthonormal set $\{e_1,\dots,e_{m}\}$. It is well known that $(V_{m},\|\cdot\|)$ and $(\mathbb{R}^{m},|\cdot|)$ are isomorphic by the natural linear transformation $T:V_{m}\to\mathbb{R}^{m}$ given by $$ v=\sum_{i=1}^{m}\xi_{i}e_{i}\to T(v)=\xi=(\xi_1,\dots,\xi_{m}) $$ which also satisfies $$ \|v\|=|T(v)|=|\xi| $$ where $|\cdot|$ and $\|\cdot\|$ denote the usual norms in $\mathbb{R}^{m}$ and $H_0^1(\Omega)$, respectively. In the next, we will use the identification $$ \xi\mapsto\sum_{i=1}^{m}\xi_{i}e_{i}=v. $$ Considering the function $F=(F_1,\dots.,F_{m}):\mathbb{R}^{m}\to\mathbb{R}^{m}$ given by $$ F_{i}(\xi)=\int_{\Omega}\nabla v\nabla e_{i}dx - \int_{\Omega}\big(h(x,|v|+\epsilon)+\lambda g(x,\nabla v)\big)e_{i}dx, $$ we have \begin{equation}\label{equa1} \begin{aligned} \langle F(\xi),\xi\rangle &= \int_{\Omega}|\nabla v|^{2}dx- \int_{\Omega}\left(h(x,|v|+\epsilon)v+\lambda g(x,\nabla v)v\right)dx \\ &\geq \|v\|^{2}- |a_1|_2|v|_2 - |a_2|_{\frac{2}{1-r_2}} |v|_2 ^{r_2+1}- \epsilon^{r_2}|a_2|_2|v|_2\\ &\quad - \frac{1}{\epsilon^{r_{3}}}|a_{3}|_2|v|_2- \lambda |a_{5}|_2|v|_2- \lambda \Big(\int_{\Omega} a_{4}^{2}|\nabla v|^{2r_{4}}dx\Big)^{1/2}|v|_2 \\ &\geq \Vert v\Vert^{2}- c_1\vert a_1\vert_{{2}}\Vert v \Vert -c_2\vert a_2\vert_{{\frac{2}{1-r_2}}}\Vert v\Vert^{r_2+1}- c_{3}\vert a_2\vert_{{2}} \Vert v \Vert \\ &\quad - c_{4} \vert a_{3}\vert_{{2}}\Vert v\Vert - c_{6} \lambda \vert a_{5}\vert_{{2}}\Vert v \Vert - \lambda c_{5}\vert a_{4}\vert_{{\frac{2}{1-r_{4}}}} \Vert v\Vert^{r_{4}+1}, \end{aligned} \end{equation} where $c_{3},c_{4}$ depend of $\epsilon$, and $c_{j}$ are independent of $m$ for $j=1,\dots,6$. Therefore, \begin{align*} \langle F(\xi),\xi\rangle &\geq |\xi|^{2}- c_1\vert a_1\vert_{{2}}|\xi| -c_2\vert a_2\vert_{{\frac{2}{1-r_2}}}|\xi|^{r_2+1}- c_{3}\vert a_2\vert_{{2}}|\xi| - c_{4} \vert a_{3}\vert_{{2}}|\xi| \\ &\quad - \lambda c_{6}\vert a_{5}\vert_{{2}}|\xi| - \lambda c_{5}\vert a_{4} \vert_{{\frac{2}{1-r_{4}}}}|\xi|^{r_{4}+1}, \end{align*} from where follows that there exist $\rho,r>0$, which are independent of $m$, such that $$ \langle F(\xi),\xi\rangle\geq r>0 \quad \text{on } |\xi|=\rho. $$ Since $F$ is a continuous functions, from Lemma \ref{le1}, for each $m\in\mathbb{N}$ there exists $\xi_{m}\in \mathbb{R}^{m}$ satisfying \begin{equation} \label{E1} F(\xi_{m})=0 \quad |\xi_{m}|\leq\rho. \end{equation} Next, we fix $v_{m} \in H_0^1(\Omega)$ such that $T(v_{m})=\xi_{m}$. Hence, $\|v_{m}\| \leq \rho $ for all $n \in \mathbb{N}$ and $$ \int_{\Omega}\nabla v_{m}\nabla \omega dx = \int_{\Omega}\big( h(x,|v_{m}|+\epsilon)\omega+\lambda g(x,\nabla v_{m})\omega\big) dx \quad \forall \omega \in V_{m}. $$ Moreover, passing to a subsequence if necessary, we can assume that there exists $v\in H_0^1(\Omega)$ such that $$ v_{m}\rightharpoonup v \quad \text{in } H_0^1(\Omega) \quad \text{and}\quad v_{m}(x)\to v(x) \quad \text{a.e in } \Omega. $$ The next claim is a key point to conclude the proof of the theorem. \begin{claim}\label{cla1} The sequence $\{v_{m}\}$ is strongly convergent to $v$ in $H^1_0(\Omega)$. \end{claim} Assuming for a moment the claim, and recalling that for $\omega\in V_{k}$ and $m\geq k$, we have the equality \begin{equation}\label{equa4} \int_{\Omega}\nabla v_{m}\nabla \omega dx = \int_{\Omega}\big( h(x,|v_{m}|+\epsilon)\omega+\lambda g(x,\nabla v_{m})\omega\big) dx. \end{equation} It follows that \begin{equation}\label{equa2} \int_{\Omega}\nabla v\nabla \omega dx = \int_{\Omega}\left(h(x,|v|+\epsilon)\omega+\lambda g(x,\nabla v)\omega\right) dx. \end{equation} Since, for each $\phi\in H_0^1(\Omega)$, there exist $\{\gamma_{i}\} \subset \mathbb{R}$ satisfying $ \phi=\sum_{i=1}^{\infty}\gamma_{i}e_{i}$, the sequence $$ \phi_{k}=\sum_{i=1}^{k}\gamma_{i}e_{i}\in V_{k}, $$ is strongly convergent to $\phi$ in $H_0^1(\Omega)$. Putting $w=\phi_{k}$ in $\eqref{equa2}$ and taking the limit as $k\to\infty$ we obtain \begin{equation}\label{equa3} \int_{\Omega}\nabla v\nabla \phi dx = \int_{\Omega}\left( h(x,|v|+\epsilon)\phi+\lambda g(x,\nabla v)\phi\right) dx. \end{equation} Thus $v$ is a weak solution of \eqref{ePe}, and by Lemma \ref{leregularidade}, $v \in C^{2}(\Omega) \cap C^1(\overline{\Omega})$. Therefore, \eqref{ePe} has a classical solution and the proof of Theorem $\ref{teo1}$ is complete. \end{proof} \begin{proof}[Proof of Claim \ref{cla1}] Using the weak convergence and the Theorem of the Dominated Convergence, it follows that \begin{gather}\label{1} \int_{\Omega}\nabla v_{m}\nabla\omega\to \int_{\Omega}\nabla v\nabla\omega, \\ \label{2} \int_{\Omega}h(x,|v_{m}|+\epsilon)\omega\to \int_{\Omega}h(x,|v|+\epsilon)\omega , \\ \label{3} \int_{\Omega}h(x,|v_{m}|+\epsilon)(v_{m}-v)\to 0, \\ \label{4} \int_{\Omega}h(x,|v|+\epsilon)(v_{m}-v)\to 0. \end{gather} From now on, for each $m \in \mathbb{N}$, we consider the function $G_{m}(x):= g(x,\nabla v_{m}(x))$. From (H3), \begin{equation} |G_{m}|_{L^{\frac{2N}{(N+2)r_{4}}}(\Omega)} \leq |a_{5}|_{L^{\frac{2N}{(N+2)r_{4}}}(\Omega)} + \Big( \int_{\Omega} a_{4}(x)^{\frac{2N}{(N+2)r_{4}}}|\nabla v_{m}|^{\frac{2N}{N+2}}dx\Big)^{\frac{(N+2)r_{4}}{2N}}. \end{equation} Using \eqref{E1} and H\"older's inequality with exponents $q =\frac{N+2}{N}$ and $p=\frac{N+2}{2}$, we get the estimate \begin{equation} |G_{m}|_{L^{\frac{2N}{(N+2)r_{4}}}(\Omega)} \leq |a_{5}|_{L^{\frac{2N}{(N+2)r_{4}}}(\Omega)} + |a_{4}|_{L^{\frac{N}{r_{4}}}(\Omega)}|\nabla v_{m}|^{r_{4}}_{L^{2} (\Omega)}\leq c_1+ c_2\rho^{r_{4}}. \end{equation} Since $L^{\frac{2N}{(N+2)r_{4}}}(\Omega)$ is reflexive, up to subsequence, there exists $G \in L^{\frac{2N}{(N+2)r_{4}}}(\Omega)$ such that $G_{m}\rightharpoonup G$ in $L^{\frac{2N}{(N+2)r_{4}}}(\Omega)$; that is, \begin{equation}\label{gm} \int_{\Omega}G_{m}\varphi dx\to\int_{\Omega}G\varphi dx \quad \forall \varphi\in L^{\theta}(\Omega) \end{equation} where $\frac{1}{\theta}+\frac{(N+2)r_{4}}{2N}=1$. Recalling that the embedding $H_0^1(\Omega)\hookrightarrow L^{\theta}(\Omega) $ is continuous, \eqref{equa4} leads to $$ \int_{\Omega}|\nabla v|^{2}dx-\int_{\Omega}h(x,|v|+\epsilon)vdx -\lambda \int_{\Omega}G(x)vdx=0. $$ On the other hand, \begin{gather*} \|v_{m}-v\|^{2}= \|v_{m}\|^{2} -\langle v_{m},v\rangle + \langle v,v_{m}-v\rangle =\|v_{m}\|^{2}-\|v\|^{2}+o_{m}(1), \\ \int_{\Omega}h(x,|v_{m}|+\epsilon)v = \int_{\Omega}h(x,|v|+\epsilon)v +o_{m}(1), \\ \int_{\Omega}G(x)v_{m}dx=\int_{\Omega}G(x)vdx + o_{m}(1). \end{gather*} From this $$ \|v_{m}-v\|^{2}= \int_{\Omega}(h(x,|v_{m}|+\epsilon)-h(x,|v| +\epsilon))v_{m}dx +\lambda \int_{\Omega}(G_{m}(x)-G(x))v_{m}dx+o_{m}(1), $$ or equivalently, \begin{align*} &\|v_{m}-v\|^{2}\\ &=\int_{\Omega}(h(x,|v_{m}|+\epsilon)-h(x,|v|+\epsilon))(v_{m}-v)dx +\lambda \int_{\Omega}(G_{m}(x)-G(x))(v_{m}-v)dx \\ &\quad + \int_{\Omega}(h(x,|v_{m}|+\epsilon)-h(x,|v|+\epsilon))v dx +\lambda \int_{\Omega}(G_{m}(x)-G(x))v dx+o_{m}(1). \end{align*} Using the weak convergence $v_{m}\rightharpoonup v$ in $H_0^1(\Omega)$, \eqref{2}-\eqref{4} and \eqref{gm}, we derive that $$ \|v_{m}-v\|^{2}\to 0. $$ This implies that $ v_{m} \to v$ in $H^1_0(\Omega)$, and the proof of Claim \ref{cla1} is complete. \end{proof} \section{Proof of Theorem \ref{TP}} In this section, we will show firstly the existence of solutions and then its regularity. \subsection*{Existence} Taking $\epsilon_{n}=1/n$ and $u_{\epsilon_{n}}=u_{n}$, it follows that \begin{equation} \begin{gathered} -\Delta u_{n}=h(x,u_{n}+ 1/n)+\lambda g(x,\nabla u_{n})\quad \text{in } \Omega\\ u_{n}>0\quad \text{in }\Omega\\ u_{n}(x)=0 \quad \text{on } \partial \Omega. \end{gathered} \label{ePn} \end{equation} By the definition of weak solution, taking $\phi=u_{n}$ as a test function in \eqref{ePn}, we have \begin{equation}\label{equ} \begin{aligned} \|u_{n}\|^{2} &\leq \Big(\int_{\Omega} a_1^2 dx\Big)^{1/2} \Big(\int_{\Omega} u_{n}^{2}dx\Big)^{1/2} +\Big(\int_{\Omega} a_2^{\frac{2}{1-r_2}}dx\Big)^{\frac{1-r_2}{2}} \Big(\int_{\Omega}u_{n}^{2}dx\Big)^{\frac{r_2+1}{2}} \\ &\quad + \Big(\int_{\Omega} a_2^{2}dx\Big)^{1/2} \Big(\int_{\Omega}u_{n}^{2}dx\Big)^{1/2} +\Big(\int_{\Omega}a_{3}^{\frac{2}{1+r_{3}}}dx\Big)^{\frac{1+r_{3}}{2}} \Big(\int_{\Omega}u_{n}^{2}dx\Big)^{\frac{1-r_{3}}{2}} \\ &\quad +\lambda \left(\int_{\Omega} a_{5}^{2}dx\right)^{1/2} \Big(\int_{\Omega} u_{n}^{2}dx\Big)^{1/2} + \lambda \Big(\int_{\Omega} a_{4}^{2}|\nabla u_{n}|^{2r_{4}}dx\Big)^{1/2} \Big(\int_{\Omega}u_{n}^{2}dx\Big)^{1/2} \\ &\leq c_1\vert a_1\vert_2\Vert u_{n}\Vert +c_2\vert a_2\vert_{\frac{2}{1-r_2}}\Vert u_{n}\|^{r_2+1}+ c_{3}\vert a_2\vert_2 \Vert u_{n} \Vert \\ &\quad + c_{4} \vert a_{3}\vert_{\frac{2}{1+r_{3}}}\Vert u_{n}\Vert^{1-r_{3}}+ \lambda c_{6}\vert a_{5}\vert_2\Vert u_{n}\Vert +\lambda c_{5}\vert a_{4}\vert_{\frac{2}{1-r_{4}}} \Vert u_{n}\Vert^{r_{4}+1} \\ &\leq \widetilde{C}(\|u_{n}\|+\|u_{n}\|^{r_2+1}+ \|u_{n}\|^{1-r_{3}}+\| u_{n}\|^{r_{4}+1}). \end{aligned} \end{equation} So there exists $K>0$ such that $\|u_{n}\|\leq K$. Up to subsequence, if necessary, we can assume that $$ u_{n}\rightharpoonup u \quad \text{in } H_0^1(\Omega), \quad u_{n}(x)\to u(x) \quad \text{a.e. in } \Omega. $$ Now, note that by (H2), $u_{n}$ is a supersolution of \begin{equation} \begin{gathered} -\Delta v=b \, v^{r_1},\quad \text{in } \Omega\\ v>0 \quad \text{in } \Omega\\ v=0 \quad \text{on } \partial \Omega. \end{gathered} \label{eB} \end{equation} Since this equation has a unique solution in $C^{2}(\Omega) \cap C(\overline{\Omega})$, which we will denote by $\Psi$, from Theorem \ref{teorsubsup}, $$ u_{n}(x) \geq \Psi(x) \quad \forall x \in \Omega,\; \forall n \in \mathbb{N} $$ and thus \begin{equation} \label{Z1} u(x) \geq \Psi(x) \quad \text{a.e in } \Omega \end{equation} from where it follows that $u(x)>0$ a.e in $\Omega$. \begin{claim} \label{C2} For each $\omega \in H_0^1(\Omega)$, we have \begin{gather}\label{5} \int_{\Omega}h(x,u_{n}+1/n)\omega dx \to\int_{\Omega}h(x,u)\omega dx,\\ \label{6} \int_{\Omega}h(x,u_{n}+1/n)(u_{n}-u) dx \to 0,\\ \label{7} \int_{\Omega}h(x,u)(u_{n}-u) dx \to 0. \end{gather} \end{claim} \begin{proof} From (H2) and \eqref{Z1}, we have \begin{gather*} |h(x,u_{n}+1/n)w|\leq c_1|w|+c_2|w||u_{n}|^{r_2} +c_{3}\frac{|w|}{\Psi^{r_{3}}}, \\ |h(x,u_{n}+1/n)(u_{n}-u)|\leq c_1|u_{n}-u|+c_2|u_{n}-u| |u_{n}|^{r_2}+c_{3}\frac{|u_{n}-u|}{\Psi^{r_{3}}}, \\ |h(x,u)(u_{n}-u)|\leq c_1|u_{n}-u|+c_2|u_{n}-u||u|^{r_2} +c_{3}\frac{|u_{n}-u|}{\Psi^{r_{3}}} \end{gather*} for some positive constants $c_{i}$ for $i=1,2,3$. From Ambrosetti, Br\'ezis and Cerami \cite{ambbrece}, the function $\Psi$ satisfies the following inequality $$ \Psi(x) \geq C \phi_1(x) \quad \forall x \in \Omega, $$ where $C$ is a positive constant and $\phi_1$ is a positive eigenfunction related to the first eingenvalue of $(-\Delta, H_0^1(\Omega))$. This way, \begin{gather}\label{8} |h(x,u_{n}+1/n)w|\leq c_1|w|+c_2|w||u_{n}|^{r_2} +c_{4}\frac{|w|}{\phi_1^{r_{3}}}, \\ \label{9} |h(x,u_{n}+1/n)(u_{n}-u)|\leq c_1|u_{n}-u| +c_2|u_{n}-u||u_{n}|^{r_2}+c_{4}\frac{|u_{n}-u|}{\phi_1^{r_{3}}},\\ \label{10} |h(x,u)(u_{n}-u)|\leq c_1|u_{n}-u|+c_2|u_{n}-u||u|^{r_2} +c_{4}\frac{|u_{n}-u|}{\phi_1^{r_{3}}}. \end{gather} From the Hardy-Sobolev inequality found in \cite{AFT} (see also \cite{Kavian} ), we have \begin{equation}\label{11} \frac{|w|}{\phi_1^{r_{3}}} \in L^1(\Omega), quad \frac{|u_{n}-u|}{\phi_1^{r_{3}}} \to 0 \quad \text{in } L^1(\Omega). \end{equation} Moreover, by using the compact Sobolev embedding, we derive that \begin{equation}\label{12} \lim_{n \to +\infty}\int_{\Omega}|u_{n}-u||u|^{r_2} =\lim_{n \to +\infty}\int_{\Omega}|u_{n}-u||u_{n}|^{r_2} = \lim_{n \to +\infty}\int_{\Omega}|u_{n}-u|=0. \end{equation} These limits together with Theorem of the Dominated Convergence imply \eqref{5}-\eqref{7}. \end{proof} Now, the Claim \ref{C2} combined with arguments explored in the proof of Claim \ref{cla1} imply $u_{n} \to u $ in $H^1_0(\Omega)$; hence, $$ \int_{\Omega}\nabla u\nabla \phi dx = \int_{\Omega}\big( h(x,u)\phi+\lambda g(x,\nabla v)\phi \big)dx \quad \forall \phi \in H_0^1(\Omega). $$ from where it follows that $u$ is a weak solution for \eqref{eP}. \begin{remark}[Regularity of the solution] \label{rmk4.1}\rm From \eqref{Z1}, it follows that $1/u$ belongs to $L_{\rm loc}^{\infty}(\Omega)$. Thus, repeating the same arguments explored in the proof of Lemma \ref{leregularidade}, it follows that $u \in C^{2}(\Omega) \cap H_0^1(\Omega)$. To conclude this study of regularity, we would like to mention the following fact: If the function $a_{3}$ given by (H2) satisfies a hypothesis of the type $$ \frac{a_{3}}{\phi_1^{r_{3}}} \in L^{\infty}(\Omega), $$ using again Regularity Theory, it follows that $u \in C^{2}(\Omega) \cap C(\overline{\Omega})$. \end{remark} \section{Proof of Theorem \ref{T2}} To prove Theorem \ref{T2}, we will work with an auxiliary problem. For each $M>M^{*}$, we define the function $$ g_{M}(t)= \begin{cases} g(t) & \text{if } 0 \leq t \leq M \\ \frac{g(M)}{M^{\beta}}t^{\beta}, & t \geq M \end{cases} $$ and the problem \begin{equation} \begin{gathered} -\Delta u=h(x,|u|)+\lambda g_{M}(|\nabla u|)\quad \text{in } \Omega\\ u>0\quad \text{in } \Omega\\ u=0 \quad \text{on } \partial \Omega. \end{gathered} \label{ePM} \end{equation} Note that if $u_{\lambda,M}$ is a solution of \eqref{ePM} satisfying \begin{equation} \label{ZZZZ} |\nabla u_{\lambda,M}(x)| \leq M \quad \forall x \in \Omega, \end{equation} then $u_{\lambda,M}$ is a solution for \eqref{eP}. From now on, our goal is to prove that there exist $M>0$ and $\lambda^{*}=\lambda^{*}(M)>0$ such that $u_{\lambda,M}$ satisfies \eqref{ZZZZ} for $\lambda \leq \lambda^{*}$. Since $h$ and $g_{M}$ satisfy the hypotheses of Theorem \ref{TP}, for all $\lambda>0$ and $M>M^{*}$, there exists a solution $u_{\lambda, M}$ of \eqref{ePM} satisfying $$ u_{\lambda, M}(x) \geq \Psi(x) \quad \forall x \in \overline{\Omega}. $$ Taking $\lambda^{*}=\frac{M^{\beta}}{g(M)}$, (H6) leads to $$ \lambda g_{M}(|\nabla u_{\lambda, M}(x)|) \leq |\nabla u_{\lambda, M}(x)|^{\beta} \quad \forall x \in \Omega,\; \forall\lambda \leq \lambda^{*}\,; $$ therefore, there exists $C>0$ such that \begin{equation} \label{W1} \|u_{\lambda, M}\| \leq C \quad \forall M>M^{*},\; \forall \lambda \leq \lambda^{*} \end{equation} where $C$ is independent of $\lambda$ and $M$. \begin{claim} \label{C1} There exists $k>0$ such that $\|u_{\lambda, M}\|_{C^1(\overline{\Omega})} \leq k$ for all $M \geq M^{*}$ and $\lambda \leq \lambda^{*}$. \end{claim} \begin{proof} By considering $$ \Phi_{\lambda, M}(x)=h(x,u_{\lambda, M}(x)) +\lambda g_{M}(|\nabla u_{\lambda,m}(x)|) $$ we have $$ |\Phi_{\lambda,M}(x)| \leq a_1(x)+ a_2(x)|u_{\lambda,M}(x)|^{r_2} + c_{7} \frac{a_{3}(x)}{\phi_1^{r_{3}}(x)} + |\nabla u_{\lambda, M}(x)|^{\beta}\quad \forall \lambda \leq \lambda^{*}. $$ From (H4), (H6) and \eqref{W1}, it follows that $\Phi_{\lambda,M} \in L^{q}(\Omega)$ for $q>\frac{N}{2}$, $ q\approx \frac{N}{2}$ and \begin{equation} \label{W2} |\Phi_{\lambda,M}|_{q} \leq C_1, \quad \forall \lambda \leq \lambda^{*},\; M \geq M^{*}. \end{equation} Since $u_{\lambda,M}$ is a solution of the problem \begin{gather*} -\Delta u(x)=\Phi_{\lambda,M}(x) \quad \text{in } \Omega\\ u(x)=0 \quad \text{on } \partial \Omega \end{gather*} by Agmon \cite[Theorem 8.2]{Ag}, $u_{\lambda,M} \in W^{2,q}(\Omega)$ and there exists $C_{3} >0$, which is independent of $u_{\lambda,M}$, such that $$ \|u_{\lambda,M}\|_{W^{2,q}(\Omega)} \leq C_{3}|\Phi_{\lambda,M}|_{q}. $$ Using the continuous embedding $W^{2,q}(\Omega) \hookrightarrow C^1(\overline{\Omega})$, we obtain \begin{equation} \label{W3} \|u_{\lambda,M}\|_{C^1(\overline{\Omega})} \leq C_{4} |\Phi_{\lambda,M}|_{q}. \end{equation} From \eqref{W2} and \eqref{W3}, there exists $k>0$ such that $$ \|u_{\lambda,M}\|_{C^1(\overline{\Omega})} \leq k \quad \forall \lambda \leq \lambda^{*} ,\; M \geq M^{*}. $$ This completes the proof of the claim. \end{proof} From Claim \ref{C1}, if $M \geq \max\{M^{*},k\}$ and $\lambda \leq \lambda^{*}$, we have $$ |\nabla u_{\lambda, M}(x)| \leq M \quad \forall x \in \Omega $$ from where it follows that $$ g_{M}(|\nabla u_{\lambda, M}(x)|) =g(|\nabla u_{\lambda, M}(x)|) \quad \forall x \in \Omega. $$ Therefore, for $\lambda \leq \lambda^{*}$ and $M \geq \max\{M^{*},k\}$ the function $u_{\lambda, M}$ is a solution for \eqref{eP}, and the proof of Theorem \ref{T2} is complete. \begin{thebibliography}{00} \bibitem{Ag} S. 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