Non-negative solutions and strong maximum principle for a resonant quasilinear problem

Abstract

We study the resonant quasilinear problem

$$\begin{aligned} -\Delta _p u = \lambda _p u^{p-1} + \lambda g(u) \text { in } \Omega ,\;\; u\ge 0 \text { in } \Omega , \;\; u_{|\partial \Omega }=0, \end{aligned}$$

where $\Omega \subset {\mathbb {R}}^N$ is a smooth, bounded domain, $\lambda _p$ is the first eigenvalue of $-\Delta _p$ in $\Omega$, and $g:[0,+\infty )\rightarrow {\mathbb {R}}$ is a continuous and subcritical term. By means of variational arguments, we prove the existence of non-negative solutions for any $\lambda >0$; positive solutions for sufficiently small $\lambda >0$. Our results generalize the ones recently obtained by different techniques in the case $p=2$.