Existence results for a class of $p$-Laplacian problems with sign-changing weight

Abstract

Consider the boundary-value problem \begin{equation} \begin{array}{c} -\Delta_p u = \lambda g(x)f(u) \quad \mbox{in}\ \Omega \nonumber\\ u=0 \quad \mbox{on}\quad \partial \Omega, \nonumber \end{array} \end{equation} where $\lambda > 0$ is a parameter, $\Omega$ is a bounded domain in $\mathbb R^N,\, N \geq 1,$ with sufficiently smooth boundary $\partial \Omega$ and $\Delta_p u:= div(|\nabla u|^{p-2}\nabla u)$ is the p-Laplacian operator for $p > 1$. Here $g$ is a $C^1$ sign-changing function that may be negative near the boundary and $f $ is a $C^1$ nondecreasing function satisfying $f(0)>0$. We discuss existence results for positive solutions when $f$ satisfies certain additional conditions. We employ the method of sub-super solutions to obtain our results.