Multiple solutions for nonlinear elliptic equations with      an asymmetric reaction term

Sophia Th. Kyritsi; Nikolaos S. Papageorgiou

doi:10.3934/dcds.2013.33.2469

Article Contents

2013, Volume 33, Issue 6: 2469-2494. Doi: 10.3934/dcds.2013.33.2469

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Multiple solutions for nonlinear elliptic equations with an asymmetric reaction term

Sophia Th. Kyritsi^1, and
Nikolaos S. Papageorgiou^2,

1.
Department of Mathematics, Hellenic Naval Academy, Piraeus 18539
2.
Department of Mathematics, National Technical University of Athens, Zografou Campus, Athens 15780

Received: January 2012

Revised: September 2012

Published: July 2013

Abstract / Introduction Related Papers Cited by

Abstract

We consider a nonlinear Dirichlet problem driven by the $p$-Laplace differential operator. We assume that the Carathéodory reaction term $f(z,x)$ exhibits an asymmetric behavior on the two semiaxes of $\mathbb{R}$. Namely, $f(z,\cdot)$ is $(p-1)$-linear near $-\infty$ and $(p-1)$-superlinear near $+\infty$, but without satisfying the well-known Ambrosetti--Rabinowitz condition (AR-condition). Combining variational methods based on critical point theory, with suitable truncation techniques and Morse theory, we show that the problem has at least three nontrivial smooth solutions, two of which have constant sign (one positive, the other negative).

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Mathematics Subject Classification: 35J25, 35J80, 58E05.

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