On a class of weighted p-Laplace equation with singular nonlinearity

TL;DR

This paper investigates the existence of solutions for a class of weighted p-Laplace equations with singular nonlinearities, establishing conditions for existence and multiplicity of solutions using variational methods.

Contribution

It introduces new existence and multiplicity results for weighted p-Laplace equations with singular nonlinearities under specific conditions.

Findings

Existence of solutions for all positive parameters ta.

At least two solutions exist in certain parameter ranges.

Solutions are obtained using variational techniques.

Abstract

This article deals with the existence of the following quasilinear degenerate singular elliptic equation \begin{equation*} (P_\la)\left\{ \begin{split} -\text{div}(w(x)|\nabla u|^{p-2}\nabla u) &= g_{\la}(u),\;u>0\; \text{in}\; \Om, u&=0 \; \text{on}\; \partial \Om, \end{split}\right. \end{equation*} where $ \Om \subset \mb R^n$ is a smooth bounded domain, $n\geq 3$, $\la>0$, $p>1$ and $w$ is a Muckenhoupt weight. Using variational techniques, for $g_{\la}(u)= \la f(u)u^{-q}$ and certain assumptions on $f$, we show existence of a solution to $(P_\la)$ for each $\la>0$. Moreover when $g_{\la}(u)= \la u^{-q}+ u^{r}$ we establish existence of atleast two solutions to $(P_\la)$ in a suitable range of the parameter $\la$. Here we assume $q\in (0,1)$ and $r \in (p-1,p^*_s-1)$.