Strong maximum principle for a sublinear elliptic problem at resonance - Giovanni Anello; Filippo Cammaroto; Luca Vilasi (Electronic Journal of Qualitative Theory of Differential Equations, 2022)
We examine the semilinear resonant problem $$ -Δ u = λ_1 u + λ g(u) \quad \text{in } Ω,\ u≥ 0 \text{ in } Ω, \ u_{|\partialΩ}=0, $$ where $Ω\subset\mathbb R^N$ is a smooth, bounded domain, $λ_1$ is the first eigenvalue of $-Δ$ in $Ω$, $λ>0$. 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 $λ>0$; the validity of the strong maximum principle for sufficiently small $λ$. The proof rests upon variational arguments.