Double resonance for Dirichlet problems with unbounded indefinite potential and combined nonlinearities

Shouchuan Hu; Nikolaos S. Papageorgiou

doi:10.3934/cpaa.2012.11.2005

Article Contents

2012, Volume 11, Issue 5: 2005-2021. Doi: 10.3934/cpaa.2012.11.2005

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Double resonance for Dirichlet problems with unbounded indefinite potential and combined nonlinearities

Shouchuan Hu^1, and
Nikolaos S. Papageorgiou^2,

1.
Department of Mathematics, Missouri State University, Springeld, MO 65804
2.
Department of Mathematics, National Technical University of Athens, Zografou Campus, Athens 15780

Received: April 30, 2011

Revised: November 30, 2011

Published: August 2012

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Abstract

We study a semilinear parametric Dirichlet equation with an indefinite and unbounded potential. The reaction is the sum of a sublinear (concave) term and of an asymptotically linear resonant term. The resonance is with respect to any nonprincipal nonnegative eigenvalue of the differential operator. Using variational methods based on the critical point theory and Morse theory (critical groups), we show that when the parameter $\lambda>0$ is small, the problem has at least three nontrivial smooth solutions.

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Mathematics Subject Classification: 35J80, 35J85, 58E05.

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