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

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.