Some remarks on semilinear resonant elliptic problems

We study existence of solutions of the semilinear elliptic problem -Delta u = a(x)u + f (u) + h(x) in Omega, u = 0 on partial derivative Omega, where Delta is the Laplace operator, a, h are L-2 (Omega)-functions with h not equal 0, a <= lambda(1) where lambda(1) is the first eigenvalue of (-Delta, H-0(1)(Omega)), f : R -> R is unbounded and continuous, and Omega subset of R-N (N >= 1) is a bounded domain with smooth boundary partial derivative Omega. We focus on "one direction resonance", namely the case f (s) = 0 for s <= 0 and inf f (s) = -infinity. No monotonicity condition is required upon s >= 0 f. Minimization arguments are exploited.