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{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, $n\ge 3$, $\la>0$, $p>1$ 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 $r\in (p-1,p_s^{\ast }-1)$.

18 pages, In this revised version the following changes are made: (1) Introduction is changed and some more references are added, (2) The assumption on $r$ is changed, (3) the additional assumption on $f$ is mentioned in (f1) and (4) the statement of Lemma 4.1 and its proof are modified