Multiple solutions with changing sign energy to a nonlinear elliptic equation

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June 2004, 3(2): 253-265. doi: 10.3934/cpaa.2004.3.253

Multiple solutions with changing sign energy to a nonlinear elliptic equation

1.	Laboratoire de Mathematiques et Applications, Universit´e de La Rochelle, 17042 La Rochelle, France

Received July 2003 Revised December 2003 Published March 2004

Abstract
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In this paper, the existence of multiple solutions to a nonlinear elliptic equation with a parameter $\lambda$ is studied. Initially, the existence of two nonnegative solutions is showed for $0 < \lambda < \hat \lambda$. The first solution has a negative energy while the energy of the second one is positive for $0 < \lambda < \lambda_0$ and negative for $\lambda_0 < \lambda < \hat \lambda$. The values $\lambda_0$ and $\hat \lambda$ are given under variational form and we show that every corresponding critical point is solution of the nonlinear elliptic problem (with a suitable multiplicative term). Finally, the existence of two classes of infinitely many solutions is showed via the Lusternik-Schnirelman theory.

Keywords: Ekeland’s principle, p-Laplacian operator, Palais-Smale condition, Lusternik-Schnirelman theory..

Mathematics Subject Classification: 35J60, 35J25, 35J6.

Citation: A. El Hamidi. Multiple solutions with changing sign energy to a nonlinear elliptic equation. Communications on Pure & Applied Analysis, 2004, 3 (2) : 253-265. doi: 10.3934/cpaa.2004.3.253

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