Eigenvalues for a Neumann Boundary Problem Involving the p ( x )‎ -Laplacian

Abstract EN

We study the existence of weak solutions to the following Neumann problem involving the p ( x ) -Laplacian operator: - Δ p ( x ) u + e ( x ) | u | p ( x ) - 2 u = λ a ( x ) f ( u ) , in Ω , ∂ u / ∂ ν = 0 , on ∂ Ω .

Under some appropriate conditions on the functions p , e , a , and f , we prove that there exists λ ¯ > 0 such that any λ ∈ ( 0 , λ ¯ ) is an eigenvalue of the above problem.

Our analysis mainly relies on variational arguments based on Ekeland’s variational principle.