Eigenvalues for A Combination Between Local and Nonlocal p-Laplacians

Abstract

Abstract In this paper we study the Dirichlet eigenvalue problem $$\begin{array}{} \displaystyle -\Delta_p u-\Delta_{J,p}u =\lambda|u|^{p-2}u \text{ in } \Omega,\quad u=0 \, \text{ in } \, \Omega^c=\mathbb{R}^N\setminus\Omega. \end{array}$$ 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 $\begin{array}{} \displaystyle \lim_{p\to\infty} \end{array}$(λ1)1/p = Λ where Λ can be characterized in terms of the geometry of Ω. We also find that the eigenfunctions converge, u∞ = $\begin{array}{} \displaystyle \lim_{p\to\infty} \end{array}$ up, and find the limit problem that is satisfied in the limit.