Annali di Matematica Pura ed Applicata (1923 -)

, Volume 195, Issue 5, pp 1771–1785

The limit as $$p\rightarrow \infty$$ in the eigenvalue problem for a system of p-Laplacians

Article

Abstract

In this paper, we study the behavior as $$p\rightarrow \infty$$ of eigenvalues and eigenfunctions of a system of p-Laplacians, that is
\begin{aligned} \left\{ \begin{array}{ll} -\Delta _p u = \lambda \alpha u^{\alpha -1} v^\beta &{}\quad \Omega , \\ -\Delta _p v = \lambda \beta u^{\alpha } v^{\beta -1} &{}\quad \Omega , \\ u=v=0, &{} \quad \partial \Omega , \end{array} \right. \end{aligned}
in a bounded smooth domain $$\Omega$$. Here $$\alpha +\beta =p$$. We assume that $$\frac{\alpha }{p} \rightarrow \Gamma$$ and $$\frac{\beta }{p} \rightarrow 1 - \Gamma$$ as $$p\rightarrow \infty$$ and we prove that for the first eigenvalue $$\lambda _{1,p}$$ we have
\begin{aligned} (\lambda _{1,p})^{1/p} \rightarrow \lambda _\infty = \frac{1}{ \max _{x \in \Omega } \hbox {dist} (x,\partial \Omega )}. \end{aligned}
Concerning the eigenfunctions $$(u_{p}, v_p)$$ associated with $$\lambda _{1,p}$$ normalized by $$\int _{\Omega } |u_p|^\alpha |v_p|^\beta =1$$, there is a uniform limit $$(u_\infty , v_\infty )$$ that is a solution to a limit minimization problem as well as a viscosity solution to
\begin{aligned} \left\{ \begin{array}{l} \min \{ -\Delta _\infty u_\infty , \, |\nabla u_\infty | - \lambda _\infty u_\infty ^{\Gamma } v_\infty ^{1-\Gamma } \} =0,\\ \min \{ -\Delta _\infty v_\infty , \, |\nabla v_\infty | - \lambda _\infty u_\infty ^{\Gamma } v_\infty ^{1-\Gamma } \} =0. \end{array} \right. \end{aligned}
In addition, we also analyze the limit PDE when we consider higher eigenvalues.

Keywords

p-Laplacian Viscosity solutions Infinity Laplacian Nonlinear eigenvalue problem

Mathematics Subject Classification

35J20 35J60 35J70

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© Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

• Denis Bonheure
• 1
Email author
• Julio D. Rossi
• 2
• Nicolas Saintier
• 2
1. 1.Département de MathématiqueUniversité libre de BruxellesBrusselsBelgium
2. 2.Departamento de Matemática, FCEyNUniversidad de Buenos AiresBuenos AiresArgentina

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