Eigenvalues for a combination between local and nonlocal $p-$Laplacians

Abstract

In this paper we study the Dirichlet eigenvalue problem $$-{\Delta }_{p}u-{\Delta }_{J,p}u=\lambda |u{|}^{p-2}u\text{ in }\Omega ,u=0\text{ in }{\Omega }^c={\mathbb{R}}^N\setminus \Omega .$$ Here ${\Delta }_{p}u$ is the standard local $p-$Laplacian, ${\Delta }_{J,p}u$ is a nonlocal, $p-$homogeneous operator of order zero and $\Omega$ is a bounded domain in ${\mathbb{R}}^N$. We show that the first eigenvalue (that is isolated and simple) satisfies $({\lambda }_1{)}^{1/p}\to \Lambda$ as $p\to \infty$ where $\Lambda$ can be characterized in terms of the geometry of $\Omega$. We also find that the eigenfunctions converge, $u_{\infty }={\lim }_{p\to \infty }{u}_p$, and find the limit problem that is satisfied in the limit.

23 pages and 3 figures