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 $$\{\begin{array}{ll}-\mathrm{div}(a_0(|\mathrm{\nabla }u{|}^{p_0})|\mathrm{\nabla }u{|}^{p_0-2}\mathrm{\nabla }u)={\displaystyle \frac{{\lambda }_0}{u^{{\beta }_0}}+f_0(u),u>0}& in\mathrm{\Omega },\\ u=0& on\partial \mathrm{\Omega },\end{array}$$ and its version for systems given by $$\{\begin{array}{ll}-\mathrm{div}(a_1(|\mathrm{\nabla }u{|}^{p_1})|\mathrm{\nabla }u{|}^{p_1-2}\mathrm{\nabla }u)={\displaystyle \frac{{\lambda }_1}{u^{{\beta }_1}}}+f_1(v)& in\mathrm{\Omega },\\ -\mathrm{div}(a_2(|\mathrm{\nabla }v{|}^{p_2})|\mathrm{\nabla }v{|}^{p_2-2}\mathrm{\nabla }v)={\displaystyle \frac{{\lambda }_2}{v^{{\beta }_2}}}+f_2(u)& in\mathrm{\Omega },\\ u,v>0& in\mathrm{\Omega },\\ u=v=0& on\partial \mathrm{\Omega },\end{array}$$ where $\mathrm{\Omega }\subset {\mathbf{R}}^N$ is bounded smooth domain with $N\ge 3$ and for $i=0,1,2$ we have $2\le p_i<N$, $0<{\beta }_i\le 1$, ${\lambda }_i>0$ and $f_i$ are continuous functions. The hypotheses on the $C^1$-functions $a_i:{\mathbf{R}}^{+}\to {\mathbf{R}}^{+}$ allow to consider a large class of quasilinear operators.