On the eigenvalue problem for the p-Laplacian operator in

In this paper, we discuss the following eigenvalue problem − D ( | D u | p − 2 D u ) = λ | u | p − 2 u , in R N , where 1 < p < N , λ ∈ R is an eigenvalue and u ≢ 0 is an eigenfunction. The main purpose of this paper is to prove that the weak form of the eigenvalue problem ∫ R N | D u | p − 2 D u D φ d x = λ ∫ R N | u | p − 2 u φ d x , ∀ φ ∈ W 1 , p ( R N ) , has only the trivial solution for any p.