On pairs of positive solutions for a class of quasilinear elliptic problems

Abstract

We prove, by using bifurcation theory, the existence of at least two positive solutions for the quasilinear problem $-\Delta_p u = f(x,u)$ in $\Omega$, $u=0$ on $\partial \Omega$, where $N>p>1$ and $\Omega$ is a smooth bounded domain in $\mathbb{R}^N$, $N\geq2,$ and the non-linearity $f$ is a locally Lipschitz continuous function, among other assumptions.