Eigenvalues for a combination between local and nonlocal p-Laplacians

Abstract

In this paper we study the Dirichlet eigenvalue problem −Δpu − ΔJ,pu = λ|u| p−2u in Ω, u = 0 in Ωc = RN \ Ω. Here Ω is a bounded domain in RN , Δpu is the standard local p-Laplacian and ΔJ,pu is a nonlocal p-homogeneous operator of order zero. We show that the first eigenvalue (that is isolated and simple) satisfies lim p→∞(λ1) 1/p = Λ where Λ can be characterized in terms of the geometry of Ω. We also find that the eigenfunctions converge, u∞ = lim p→∞ up, and find the limit problem that is satisfied in the limit. MSC 2010: Primary 35R11; Secondary 35J92, 35P30, 47G20