| Summary: | Abstract This paper uses the Galerkin method to investigate the existence of positive solution to a class of singular elliptic problems given by { − 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 , $$\begin{aligned} \textstyle\begin{cases} -\Delta u= \displaystyle \frac {\lambda _{0}}{u^{\beta _{0}}} + \Lambda _{0} |\nabla u|^{\gamma _{0}}+ \frac{f_{0}(u)}{|x|^{\alpha _{0}}}+ h_{0}(x), \ \ u>0 \ \ \text{in} \ \Omega , \\ u=0 \ \text{on} \ \ \partial \Omega , \end{cases}\displaystyle \end{aligned}$$ where Ω ⊂ R 2 $\Omega \subset \mathbb{R}^{2}$ is a bounded smooth domain, 0 < β 0 $0<\beta _{0}$ , γ 0 ≤ 1 $\gamma _{0} \leq 1$ , α 0 ∈ [ 0 , 2 ) $\alpha _{0} \in [0,2)$ , h 0 ( x ) ≥ 0 $h_{0}(x)\geq 0$ , h 0 ≠ 0 $h_{0}\neq 0$ , h 0 ∈ L ∞ ( Ω ) $h_{0}\in L^{\infty}(\Omega )$ , 0 < ∥ h 0 ∥ ∞ < λ 0 < Λ 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 ( ϒ ‾ s 2 ) $\exp (\overline{\Upsilon}s^{2})$ as | s | → ∞ $|s| \to \infty $ , for some ϒ ‾ > 0 $\overline{\Upsilon}>0$ .
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