Gabriel Lopez Garza & Adolfo J. Rumbos
We prove the existence of weak solutions for the semilinear elliptic problem
where , , is a continuous bounded function, and , . We assume that in the case of resonance and that and 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 of previous results in bounded domains by Ahmad, Lazer, and Paul , for the case of resonance, and by Silva  in the strong resonance case.
Submitted June 3, 2003. Published December 16, 2003.
Math Subject Classifications: 35J20.
Key Words: Resonance, strong resonance, concentration-compactness.
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| Gabriel Lopez Garza |
Dept. of Math., Claremont Graduate University
Claremont California 91711, USA
| Adolfo J. Rumbos |
Department of Mathematics, Pomona College
Claremont, California 91711, USA
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