Multiple non-trivial solutions of the Neumann problem for p-Laplacian systems
We obtain multiple non-trivial solutions of the Neumann problem for p-Laplacian systems using Morse theory.
THE MORSE INDEX OF A SADDLE POINT
This paper is devoted to clarifying the relationship between the classical Morse theory andthe Mountain Pass Lemma via the local linking concept.It is shown that for a C~1-function f with alocal linking,the m-thcritical group is nontrivial,where m is the Morse index.Combined with thebehavior of f at infinity,this result can be used to offer the existence of nontrivial critical points.
On eigenvalue problems of -Laplacian with Neumann boundary conditions
We study the nonlinear eigenvalue problem −Δ p u=λm(x)|u| p−2 u in Ω. ∂u/∂n=0 on ∂Ω, where p>1. For ∫ Ω m(x)<0, we prove that the first positive eigenvalue λ 1 exists and is simple and unique, in the sense that it is the only eigenvalue with a positive eigenfunction. In the case ∫ Ω m(x)=0, we prove that λ 0 =0 is the only eigenvalue with a positive eigenfunction