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Asymptotic behaviour as p → ∞ of least energy solutions of a (p, q(p))-Laplacian problem

Part of: Partial differential equations Linear function spaces and their duals Elliptic equations and systems Representations of solutions

Published online by Cambridge University Press:  17 January 2019

C. O. Alves ,
G. Ercole  and
G. A. Pereira
C. O. Alves
Affiliation:
Universidade Federal de Campina Grande, Campina Grande, PB 58.109-970, Brazil (coalves@mat.ufcg.edu.br)
G. Ercole
Affiliation:
Universidade Federal de Minas Gerais, Belo Horizonte, MG 30.123-970, Brazil (grey@mat.ufmg.br; gilbertoapereira@yahoo.com.br)
G. A. Pereira
Affiliation:
Universidade Federal de Minas Gerais, Belo Horizonte, MG 30.123-970, Brazil (grey@mat.ufmg.br; gilbertoapereira@yahoo.com.br)
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Abstract

We study the asymptotic behaviour, as p → ∞, of the least energy solutions of the problem

$$\left\{ {\matrix{ {-(\Delta _p + \Delta _{q(p)})u = \lambda _p \vert u(x_u) \vert ^{p-2}u(x_u)\delta _{x_u}} & {{\rm in}} & \Omega \cr {u = 0} \hfill \hfill \hfill & {{\rm on}} & {\partial \Omega ,} \cr } } \right.$$
where xu is the (unique) maximum point of |u|, $\delta _{x_{u}}$ is the Dirac delta distribution supported at xu,
$$\mathop {\lim }\limits_{p\to \infty } \displaystyle{{q(p)} \over p} = Q\in \left\{ {\matrix{ {(0,1)} & {{\rm if}} & {N < q(p) < p} \cr {(1,\infty )} & {{\rm if}} & {N < p < q(p)} \cr } } \right.$$
and λp > 0 is such that
$$\min \left\{ {\displaystyle{{{\rm \Vert }\nabla u{\rm \Vert }_\infty } \over {{\rm \Vert }u{\rm \Vert }_\infty }}:0 \ne u\in W^{1,\infty }(\Omega )\cap C_0(\bar{\Omega })} \right\} \les \mathop {\lim }\limits_{p\to \infty } (\lambda _p)^{1/p} < \infty .$$

Keywords

Asymptotic behaviourdirac deltainfinity LaplacianNehari setviscosity solution

MSC classification

Primary: 35B40: Asymptotic behavior of solutions
Secondary: 35D40: Viscosity solutions 35J20: Variational methods for second-order elliptic equations 35J92: Quasilinear elliptic equations with $p$-Laplacian 46E35: Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 149 , Issue 6 , December 2019 , pp. 1493 - 1522
Copyright
Copyright © Royal Society of Edinburgh 2019 

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