Multiple solutions with changing sign energy to a nonlinear elliptic equation

Pages: 253 - 265, Volume 3, Issue 2, June 2004

doi:10.3934/cpaa.2004.3.253       Abstract        Full Text (222.3K)       Related Articles

A. El Hamidi - Laboratoire de Mathematiques et Applications, Universit´e de La Rochelle, 17042 La Rochelle, France (email)

Abstract: 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, 35J65.

Received: July 2003;      Revised: December 2003;      Published: March 2004.