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Nontrivial Solutions for Semilinear Elliptic Problems with Resonance

where Ω ⊂ R (N ≥ 1) is a bounded smooth domain, λk is an eigenvalue of the problem −Δu = λu in Ω, u = 0 on ∂Ω, and f : Ω × R → R is a Carathéodory function. If f(x, 0) = 0 for a.e. x ∈ Ω the constant u = 0 is a trivial solution of the problem (P). In this case, the key point is proving the existence of nontrivial solutions for (P). For this purpose, we need to introduce some conditions on the behaviors of the perturbed function f(x, t) or its primitive F (x, t) = ∫ t 0 f(x, s) ds near infinity and near zero. There have been many papers concerning problem (P) at resonance; see for example [1]-[8] and the references therein. The authors of [1] proved an existence result in the case where f is uniformly bounded and the primitive of the function f , satisfies ∫