Some Remarks on Semilinear Resonant Elliptic Problems

Abstract

We study existence of solutions of the semilinear elliptic problem

$$ - \Delta u = a\left( x \right)u + f\left( u \right) + h\left( x \right)in\Omega ,u = 0on\partial \Omega $$

, where Δ is the Laplace operator, a, h are L ²(Ω)-functions with h≠0, a≤λ ₁ where λ ₁ is the first eigenvalue of (−Δ, H ¹₀ (Ω)), f : R → R is unbounded and continuous, and Ω ⊂ R ^N (N≥1) is a bounded domain with smooth boundary ∂Ω. We focus on “one direction resonance”, namely the case f(s)=0 for s≤0 and $\mathop {\inf }\limits_{s \geqslant 0}$ f(s)=−∞. No monotonicity condition is required upon f. Minimization arguments are exploited.

Research partially supported by CNPq/PQ, PADCT/UFG 620039/2004-8, PRONEX/UnB.