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

Maya Chhetri, Pavel Drabek, Ratnasingham Shivaji

Abstract

We consider an eigenvalue problem of the form \begin{equation*} \left\{\begin{array}{rclll} -Δ_{p} u -Δ_{q} u&=& λK(x)|u|^{p-2}u &\mbox{ in } Ω^e u&=&0\qquad \quad &\mbox{ on } \partial Ω u(x) &\to& 0 &\mbox{ as } |x| \to \infty\,, \end{array}\right. \end{equation*} where $Ω^e$ is the exterior of a simply connected, bounded domain $Ω$ in $\mathbb{R}^N$, $p, q \in (1, N)$ with $p \neq q$, $0 < K \in L^{\infty}(Ω^e) \cap L^{\frac{N}{p}}(Ω^e)$, and $λ\in \mathbb{R}$. 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 $λ$. We use the {\em fibering method} of S.~I. Pohozaev to prove our results.

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

Abstract

We consider an eigenvalue problem of the form \begin{equation*} \left\{\begin{array}{rclll} -Δ_{p} u -Δ_{q} u&=& λK(x)|u|^{p-2}u &\mbox{ in } Ω^e u&=&0\qquad \quad &\mbox{ on } \partial Ω u(x) &\to& 0 &\mbox{ as } |x| \to \infty\,, \end{array}\right. \end{equation*} where is the exterior of a simply connected, bounded domain in , with , , and . 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 . We use the {\em fibering method} of S.~I. Pohozaev to prove our results.

Paper Structure

This paper contains 7 sections, 7 theorems, 76 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Fibering method
  3. The case: $p < q$
  4. The case: $p> q$
  5. Asymptotics behavior of solutions
  6. Regularity of solutions
  7. Final Remarks

Key Result

Theorem 1.1

Equation pde admits a nontrivial weak solution $u_{\lambda} \in X \cap L^{\infty}(\Omega^e)$ such that $u_{\lambda} >0$ in $\Omega^e$ and $\lim\limits_{|x| \to \infty}u_{\lambda}(x)=0$ (uniformly) if and only if $\lambda \in (\lambda_1(p), \, +\infty)$. If $\partial \Omega$ is of class $C^2$, then $

Figures (2)

  • Figure 1: Qualitative behaviors of $J_{\lambda}(u_{\lambda})$ and $\|u_{\lambda}\|_q$ as functions of $\lambda$ when $p<q$.
  • Figure 2: Qualitative behaviors of $J_{\lambda}(u_{\lambda})$ and $\|u_{\lambda}\|_q$ as functions of $\lambda$ when $p>q$.

Theorems & Definitions (14)

  • Definition 1.1
  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 3.1
  • proof
  • Lemma 4.1
  • proof
  • ...and 4 more