Multiplicity results for a quasilinear equation with singular nonlinearity

TL;DR

This paper proves the existence of at least two solutions for a singular quasilinear PDE with specific boundary conditions in a convex domain, under certain parameter constraints.

Contribution

It establishes multiplicity results for solutions to a class of singular quasilinear equations with nonlinearities, extending previous results by considering broader parameter ranges.

Findings

Existence of at least two solutions for the PDE under certain conditions.

Solutions have specific regularity properties, with their powers in Sobolev spaces.

Results hold for all positive  and  within specified bounds.

Abstract

For an open, bounded domain $\Om$ in $\mathbb{R}^N$ which is strictly convex with $C^2$ boundary, we show that there exists a $\land>0$ such that the singular quasilinear problem \begin{eqnarray*} &-\delp u =\cfrac{\lambda}{u^{\del}}+u^q\,\,\mbox{in}\,\,\Om\\ &u=0\,\,\mbox{on}\,\,\partial\Om;\, \,\,u>0\,\,\mbox{in}\,\,\Om \end{eqnarray*} admits atleast two solution $ u$ and $v$ in $W^{1,p}_{loc}(\Om)\cap L^{\infty}(\Om)$ for any $\del>0$ and $0<\lam<\land$ provided $1<p<N$ and $p-1<q<\frac{p(N-1)}{N-p}-1$.\\ Moreover the solutions $u$ and $v$ are such that $u^{\alp}$ and $v^{\alp}$ are in $W^{1,p}_0(\Om)$ for some $\alp>0$.