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 $$\begin{aligned} \textstyle\begin{cases} -\Delta u= \displaystyle \frac {\lambda _{0}}{u^{\beta _{0}}} + \Lambda _{0} |\nabla u|^{\gamma _{0}}+ \frac{f_{0}(u)}{|x|^{\alpha _{0}}}+ h_{0}(x), \ \ u>0 \ \ \text{in} \ \Omega , \\ u=0 \ \text{on} \ \ \partial \Omega , \end{cases}\displaystyle \end{aligned}$$ { − Δ u = λ 0 u β 0 + Λ 0 | ∇ u | γ 0 + f 0 ( u ) | x | α 0 + h 0 ( x ) , u > 0 in Ω , u = 0 on ∂ Ω , where $\Omega \subset \mathbb{R}^{2}$ Ω ⊂ R 2 is a bounded smooth domain, $0<\beta _{0}$ 0 < β 0 , $\gamma _{0} \leq 1$ γ 0 ≤ 1 , $\alpha _{0} \in [0,2)$ α 0 ∈ [ 0 , 2 ) , $h_{0}(x)\geq 0$ h 0 ( x ) ≥ 0 , $h_{0}\neq 0$ h 0 ≠ 0 , $h_{0}\in L^{\infty}(\Omega )$ h 0 ∈ L ∞ ( Ω ) , $0<\|h_{0}\|_{\infty} < \lambda _{0} < \Lambda _{0}$ 0 < ∥ h 0 ∥ ∞ < λ 0 < Λ 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 $\exp (\overline{\Upsilon}s^{2})$ exp ( ϒ ‾ s 2 ) as $|s| \to \infty $ | s | → ∞ , for some $\overline{\Upsilon}>0$ ϒ ‾ > 0 .