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

Abstract

This paper uses the Galerkin method to investigate the existence of positive solution to a class of singular elliptic problems given by $$\begin{aligned} \textstyle\begin{cases} -\Delta u= \displaystyle \frac {\lambda _{0}}{u^{\beta _{0}}} + \Lambda _{0} |\nabla u|^{\gamma _{0}}+ \frac{f_{0}(u)}{|x|^{\alpha _{0}}}+ h_{0}(x), \ \ u>0 \ \ \text{in} \ \Omega , \\ u=0 \ \text{on} \ \ \partial \Omega , \end{cases}\displaystyle \end{aligned}$$ where $\Omega \subset \mathbb{R}^{2}$ is a bounded smooth domain, $0<\beta _{0}$ , $\gamma _{0} \leq 1$ , $\alpha _{0} \in [0,2)$ , $h_{0}(x)\geq 0$ , $h_{0}\neq 0$ , $h_{0}\in L^{\infty}(\Omega )$ , $0<\|h_{0}\|_{\infty} < \lambda _{0} < \Lambda _{0}$ , and $f_{0}$ are continuous functions. More precisely, $f_{0}$ has a critical exponential growth, that is, the nonlinearity behaves like $\exp (\overline{\Upsilon}s^{2})$ as $|s| \to \infty $ , for some $\overline{\Upsilon}>0$ .