On a Class of Weighted p-Laplace Equation with Singular Nonlinearity

Abstract

This article deals with the existence of the following quasilinear degenerate singular elliptic equation:

$$\begin{aligned} (P_\lambda )\left\{ \begin{aligned} -\text {div}(w(x)|\nabla u|^{p-2}\nabla u)&= g_{\lambda }(u),\;u>0\; \text {in}\; \Omega ,\\ u&=0 \quad \text {on}\quad \partial \Omega , \end{aligned}\right. \end{aligned}$$

where $\Omega \subset {\mathbb {R}}^n$ is a smooth bounded domain, $n\ge 3$, $\lambda >0$, $p>1$, and w is a Muckenhoupt weight. Using variational techniques, for $g_{\lambda }(u)= \lambda f(u)u^{-q}$ and certain assumptions on f, we show existence of a solution to $(P_\lambda )$ for each $\lambda >0$. Moreover, when $g_{\lambda }(u)= \lambda u^{-q}+ u^{r}$, we establish existence of at least two solutions to $(P_\lambda )$ in a suitable range of the parameter $\lambda$. Here, we assume $q\in (0,1)$ and $r \in (p-1,p^*_s-1)$.