On an asymptotically linear singular boundary value problems

Abstract

We prove the existence of positive solutions for the singular boundary value problems $$ \begin{cases} \displaystyle -\Delta u=\frac{p(x)}{u^{\beta }}+\lambda f(u) & \text{in }\Omega , \\ u=0 &\text{on }\partial \Omega , \end{cases} $$ where $\Omega $ is a bounded domain in $\mathbb{R}^n$ with smooth boundary $\partial \Omega$, $0< \beta < 1$, $\lambda > 0$ is a small parameter, $f\colon (0,\infty )\rightarrow \mathbb{R}$ is asymptotically linear at $\infty$ and is possibly singular at $0$.