Electronic Journal of Qualitative Theory of Differential Equations (Jul 2022)
Strong maximum principle for a sublinear elliptic problem at resonance
Abstract
We examine the semilinear resonant problem $$ -\Delta u = \lambda_1 u + \lambda g(u) \quad \text{in } \Omega,\ u\geq 0 \text{ in } \Omega, \ u_{|\partial\Omega}=0, $$ where $\Omega\subset\mathbb R^N$ is a smooth, bounded domain, $\lambda_1$ is the first eigenvalue of $-\Delta$ in $\Omega$, $\lambda>0$. Inspired by a previous result in literature involving power-type nonlinearities, we consider here a generic sublinear term $g$ and single out conditions to ensure: the existence of solutions for all $\lambda>0$; the validity of the strong maximum principle for sufficiently small $\lambda$. The proof rests upon variational arguments.
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