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

We study the resonant quasilinear problem $-\Delta_p u = \lambda_p u^{p-1} + \lambda g(u) in \Omega, u \ge 0 in \Omega, u_{\partial\Omega} = 0,$ where $\Omega\subset\mathbb{R}^n$ is a smooth, bounded domain, $λ_p$ is the first eigenvalue of $-\Delta_p$ in $\Omega$, and $g : [0, +\infty)\to\mathbb{R}$ 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 $p = 2$.