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

This paper uses the Galerkin method to investigate the existence of positive solution to a class of singular elliptic problems given by $\{\begin{array}{c}-\mathrm{\Delta }u=\frac{{\lambda }_0}{u^{{\beta }_0}}+{\mathrm{\Lambda }}_0{|\mathrm{\nabla }u|}^{{\gamma }_0}+\frac{f_0(u)}{{|x|}^{{\alpha }_0}}+h_0(x),u>0\text{in}\mathrm{\Omega },\hfill \\ u=0\text{on}\partial \mathrm{\Omega },\hfill \end{array}$ \(\begin{aligned} \textstyle\begin{cases} -\Delta u= \displaystyle \frac {\lambda _{0}}{u^{\beta _{0}}} + \Lambda _{0} |\nabla u|^{\gamma _{0}}+ \frac{f_{0}(u)}{|x|^{\alpha _{0}}}+ h_{0}(x), \ \ u>0 \ \ \text{in} \ \Omega , \\ u=0 \ \text{on} \ \ \partial \Omega , \end{cases}\displaystyle \end{aligned}\) where $\mathrm{\Omega }\subset {\mathbb{R}}^2$ $\Omega \subset \mathbb{R}^{2}$ is a bounded smooth domain, $0<{\beta }_0$ $0<\beta _{0}$ , ${\gamma }_0\le 1$ $\gamma _{0} \leq 1$ , ${\alpha }_0\in [0,2)$ $\alpha _{0} \in [0,2)$ , $h_0(x)\ge 0$ $h_{0}(x)\geq 0$ , $h_0\ne 0$ $h_{0}\neq 0$ , $h_0\in L^{\mathrm{\infty }}(\mathrm{\Omega })$ $h_{0}\in L^{\infty}(\Omega )$ , $0<{\parallel h_0\parallel }_{\mathrm{\infty }}<{\lambda }_0<{\mathrm{\Lambda }}_0$ $0<\|h_{0}\|_{\infty} < \lambda _{0} < \Lambda _{0}$ , and $f_0$ $f_{0}$ are continuous functions. More precisely, $f_0$ $f_{0}$ has a critical exponential growth, that is, the nonlinearity behaves like $exp(\bar{\mathrm{\Upsilon }}{s}^2)$ $\exp (\overline{\Upsilon}s^{2})$ as $|s|\to \mathrm{\infty }$ $|s| \to \infty $ , for some $\bar{\mathrm{\Upsilon }}>0$ $\overline{\Upsilon}>0$ .