Existence of positive solutions for a class of singular and quasilinear elliptic problems with critical exponential growth

Abstract

In this paper we use Galerkin method to investigate the existence of positive solution for a class of singular and quasilinear elliptic problems given by \[\begin{cases}-\operatorname{div}(a_0(|\nabla u|^{p_0})|\nabla u|^{p_{0}-2}\nabla u)= \displaystyle\frac {\lambda_0}{u^{\beta_0}} + f_0(u),\ u>0 &in\ \Omega,\\ u=0 &on\ \partial\Omega,\end{cases}\] and its version for systems given by \[\begin{cases}-\operatorname{div}(a_1(\vert\nabla u\vert^{p_1})\ \vert \nabla u\vert ^{p_1-2}\ \nabla u)=\dfrac{\lambda_1}{u^{\beta_1}}+f_1(v) &in\ \Omega,\\ -\operatorname{div}(a_2(\vert\nabla v\vert^{p_2})\ \vert \nabla v\vert ^{p_2-2}\ \nabla v)=\dfrac{\lambda_2}{v^{\beta_2}}+f_2(u) &in\ \Omega,\\ u,v>0 &in\ \Omega,\\ u=v=0 &on\ \partial\Omega,\end{cases}\] where \(\Omega\subset\mathbf{R}^{N}\) is bounded smooth domain with \(N\geq 3\) and for \(i=0,1,2\) we have \(2 \leq p_i < N\), \(0<\beta_i \leq 1\), \(\lambda_i>0\) and \(f_i\) are continuous functions. The hypotheses on the \(C^1\)-functions \(a_i\colon \mathbf{R}^+\rightarrow \mathbf{R}^+\) allow to consider a large class of quasilinear operators.