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

• Denis Bonheure
• Julio D. Rossi
• Nicolas Saintier
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

## Notes

### Acknowledgments

JDR was partially supported by MEC MTM2010-18128 and MTM2011-27998 (Spain). Part of this work was done during a visit of JDR to Univ. libre de Bruxelles. He wants to thank for the very nice and stimulating atmosphere found there. DB is supported by INRIA— Team MEPHYSTO, MIS F.4508.14 (FNRS), PDR T.1110.14F (FNRS) & ARC AUWB-2012-12/17-ULB1-IAPAS.

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## 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