Leandro Recova, Adolfo Rumbos
Abstract:
We study the existence and multiplicity of solutions of the problem
where
is a smooth bounded domain in
,
denotes the negative part of
,
is
the first eigenvalue of the N-dimensional Laplacian with Dirichlet
boundary conditions in
,
and
is a continuous function with
for all
.
We assume that the nonlinearity g(x,s) has a strong resonant behavior
for large negative values of s and is superlinear, but subcritical,
for large positive values of s. Because of the lack of compactness
in this kind of problem, we establish conditions under which the associated
energy functional satisfies the Palais-Smale condition. We prove the
existence of three nontrivial solutions of the problem as a
consequence of Ekeland's Variational Principle and a variant of the mountain
pass theorem due to Pucci and Serrin [14].
Submitted February 19, 2017. Published June 23, 2017.
Math Subject Classifications: 35J20.
Key Words: Strong resonance; Palais-Smale condition; Ekeland's principle.
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Leandro L. Recova Ericsson Inc., Irvine, California 92609, USA email: leandro.recova@ericsson.com | |
Adolfo J. Rumbos Department of Mathematics Pomona College Claremont, California 91711, USA email: arumbos@pomona.edu |
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