JournalsaihpcVol. 25, No. 1pp. 181–200

Solutions of an elliptic system with a nearly critical exponent

  • I.A. Guerra

    Departamento de Matemáticas y Ciencia de la Computación, Facultad de Ciencia, Universidad de Santiago de Chile, Casilla 307, Correo 2, Santiago, Chile
Solutions of an elliptic system with a nearly critical exponent cover
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Abstract

Consider the problem

Δuɛ=vɛp,vɛ>0inΩ,−\mathrm{\Delta }u_{ɛ} = v_{ɛ}^{p},\:v_{ɛ} > 0\:\text{in}\:\Omega ,
Δvɛ=uɛqɛ,uɛ>0inΩ,−\mathrm{\Delta }v_{ɛ} = u_{ɛ}^{q_{ɛ}},\:u_{ɛ} > 0\:\text{in}\:\Omega ,
uɛ=vɛ=0onΩ,u_{ɛ} = v_{ɛ} = 0\:\text{on}\:\partial \Omega ,

where Ω is a bounded convex domain in RN\mathbb{R}^{N}, N>2N > 2, with smooth boundary ∂Ω. Here p,qɛ>0p,q_{ɛ} > 0, and

ɛ:=Np+1+Nqɛ+1(N2).ɛ: = \frac{N}{p + 1} + \frac{N}{q_{ɛ} + 1}−(N−2).

This problem has positive solutions for ɛ>0ɛ > 0 (with pqɛ>1pq_{ɛ} > 1) and no non-trivial solution for ɛ0ɛ⩽0. We study the asymptotic behavior of least energy solutions as ɛ0+ɛ\rightarrow 0^{ + }. These solutions are shown to blow-up at exactly one point, and the location of this point is characterized. In addition, the shape and exact rates for blowing up are given.

Résumé

Considérons le problème

Δuɛ=vɛp,vɛ>0enΩ,−\mathrm{\Delta }u_{ɛ} = v_{ɛ}^{p},\:v_{ɛ} > 0\:\text{en}\:\Omega ,
Δvɛ=uɛqɛ,uɛ>0enΩ,−\mathrm{\Delta }v_{ɛ} = u_{ɛ}^{q_{ɛ}},\:u_{ɛ} > 0\:\text{en}\:\Omega ,
uɛ=vɛ=0surΩ,u_{ɛ} = v_{ɛ} = 0\:\text{sur}\:\partial \Omega ,

Ω est un domaine convexe et borné de RN\mathbb{R}^{N}, N>2N > 2, avec la frontière régulière ∂Ω. Ici p,qɛ>0p,q_{ɛ} > 0, et

ɛ:=Np+1+Nqɛ+1(N2).ɛ: = \frac{N}{p + 1} + \frac{N}{q_{ɛ} + 1}−(N−2).

Ce problème a des solutions positives pour ɛ>0ɛ > 0 (avec pqɛ>1pq_{ɛ} > 1) et aucune solution non-triviale pour ɛ0ɛ⩽0. Nous étudions le comportement asymptotique de solutions d'énergie minimale quand ɛ0+ɛ\rightarrow 0^{ + }. Ces solutions explosent en un seul point, et la position de ce point est caracterisée. De plus, le profil et les vitesses exactes d'explosion sont donnés.

Cite this article

I.A. Guerra, Solutions of an elliptic system with a nearly critical exponent. Ann. Inst. H. Poincaré Anal. Non Linéaire 25 (2008), no. 1, pp. 181–200

DOI 10.1016/J.ANIHPC.2006.11.008