Strong maximum principle for a sublinear elliptic problem at resonance

We examine the semilinear resonant problem −∆u = λ1 u + λg(u) in Ω, u ≥ 0 in Ω, u|∂Ω = 0, where Ω ⊂ RN is a smooth, bounded domain, λ1 is the first eigenvalue of −∆ 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.