Electron. J. Diff. Eqns., Vol. 2003(2003), No. 124, pp. 122.
Resonance and strong resonance for semilinear elliptic equations
in
Gabriel Lopez Garza & Adolfo J. Rumbos
Abstract:
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 PalaisSmale
condition holds for the functional associated with the semilinear problem
using the concentrationcompactness lemma of Lions. Then we prove the
existence of weak solutions by applying the saddle point theorem of
Rabinowitz for the cases of nonresonance 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 [2], for the case
of resonance, and by Silva [15] in the strong resonance case.
Submitted June 3, 2003. Published December 16, 2003.
Math Subject Classifications: 35J20.
Key Words: Resonance, strong resonance, concentrationcompactness.
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Gabriel Lopez Garza
Dept. of Math., Claremont Graduate University
Claremont California 91711, USA
email: Gabriel.Lopez@cgu.edu 

Adolfo J. Rumbos
Department of Mathematics, Pomona College
Claremont, California 91711, USA
email: arumbos@pomona.edu 
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