On principal eigenpairs for the (p,q)-Laplacian in exterior domain

TL;DR

This paper investigates the principal eigenpairs of a nonlinear (p,q)-Laplacian problem in exterior domains, establishing existence, regularity, positivity, and asymptotic behavior of eigenfunctions using the fibering method.

Contribution

It proves the existence of an unbounded set of principal eigenvalues and eigenfunctions for the (p,q)-Laplacian in exterior domains, with detailed properties.

Findings

Existence of an unbounded set of principal eigenvalues.

Eigenfunctions exhibit regularity, positivity, and specific asymptotic profiles.

Use of the fibering method to establish these results.

Abstract

We consider an eigenvalue problem of the form \begin{equation*} \left\{\begin{array}{rclll} -\Delta_{p} u -\Delta_{q} u&=& \lambda K(x)|u|^{p-2}u & \mbox{ in } \Omega^e u&=&0\qquad \quad &\mbox{ on } \partial \Omega u(x) &\to& 0 &\mbox{ as } |x| \to \infty\,, \end{array}\right. \end{equation*} where $\Omega^e$ is the exterior of a simply connected, bounded domain $\Omega$ in $\mathbb{R}^N$, $p, q \in (1, N)$ with $p \neq q$, $0 < K \in L^{\infty}(\Omega^e) \cap L^{\frac{N}{p}}(\Omega^e)$, and $\lambda \in \mathbb{R}$. We establish the existence of an unbounded set of the principal eigenvalues and corresponding eigenfunctions. Moreover, we establish the regularity, positivity and the asymptotic profiles of these eigenfunctions with respect to the eigenvalue parameter $\lambda$. We use the {\em fibering method} of S.~I. Pohozaev to prove our results.