Abstract and Applied Analysis
Volume 2 (1997), Issue 3-4, Pages 185-195
doi:10.1155/S108533759700033X

A result on the bifurcation from the principal eigenvalue of the Ap-Laplacian

P. Drábek,1 A. Elkhalil,2 and A. Touzani2

1Department of Mathematics, University of West Bohemia, P.O. Box 314, 306 14 Pilsen, Czech Republic
2Département des Mathématiques, Faculté des Sciences Dhar-Mahraz, B. P. 1796, Fes–Atlas, Fes, Morocco

Received 1 July 1997

Copyright © 1997 P. Drábek et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We study the following bifurcation problem in any bounded domain Ω in ℝN: {Apu:=−∑i,j=1N∂∂xi[(∑m,k=1Namk(x)∂u∂xm∂u∂xk)p−22aij(x)∂u∂xj]=                         λg(x)|u|p−2u+f(x,u,λ),u∈W01,p(Ω).. We prove that the principal eigenvalue λ1 of the eigenvalue problem {Apu=λg(x)|u|p−2u,u∈W01,p(Ω), is a bifurcation point of the problem mentioned above.