RESONANCE AND STRONG RESONANCE FOR SEMILINEAR ELLIPTIC EQUATIONS IN RN

We prove the existence of weak solutions for the semilinear elliptic problem [GRAPHICS] Where lambda is an element of R, f is an element of L2N/(N+ 2), g : R -> R is a continuous bounded function, and h is an element of L-N/2 boolean AND L-alpha, alpha > N/2. We assume that alpha is an element of L2N/( N+ 2) boolean AND L-infinity in the case of resonance and that a is an element of L-1 boolean AND L-infinity and f = 0 for the case of strong resonance. We prove first that the Palais-Smale condition holds for the functional associated with the semilinear problem using the concentration-compactness lemma of Lions. Then we prove the existence of weak solutions by applying the saddle point theorem of Rabinowitz for the cases of non-resonance and resonance, and a linking theorem of Silva in the case of strong resonance. The main theorems in this paper constitute an extension to R-N of previous results in bounded domains by Ahmad, Lazer, and Paul [2], for the case of resonance, and by Silva [15] in the strong resonance case.