Solutions of an elliptic system with a nearly critical exponent

I.A. Guerra

doi:10.1016/j.anihpc.2006.11.008

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

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Abstract

Consider the problem

- Δ u_{ɛ} = v_{ɛ}^{p}, v_{ɛ} > 0 in Ω,

- Δ v_{ɛ} = u_{ɛ}^{q_{ɛ}}, u_{ɛ} > 0 in Ω,

u_{ɛ} = v_{ɛ} = 0 on \partial Ω,

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

ɛ := \frac{N}{p + 1} + \frac{N}{q _{ɛ} + 1} - (N - 2) .

This problem has positive solutions for $ɛ > 0$ (with $p q_{ɛ} > 1$ ) and no non-trivial solution for $ɛ ⩽ 0$ . We study the asymptotic behavior of least energy solutions as $ɛ \to 0^{+}$ . 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

- Δ u_{ɛ} = v_{ɛ}^{p}, v_{ɛ} > 0 en Ω,

- Δ v_{ɛ} = u_{ɛ}^{q_{ɛ}}, u_{ɛ} > 0 en Ω,

u_{ɛ} = v_{ɛ} = 0 sur \partial Ω,

où $Ω$ est un domaine convexe et borné de $R^{N}$ , $N > 2$ , avec la frontière régulière $\partial Ω$ . Ici $p, q_{ɛ} > 0$ , et

ɛ := \frac{N}{p + 1} + \frac{N}{q _{ɛ} + 1} - (N - 2) .

Ce problème a des solutions positives pour $ɛ > 0$ (avec $p q_{ɛ} > 1$ ) et aucune solution non-triviale pour $ɛ ⩽ 0$ . Nous étudions le comportement asymptotique de solutions d'énergie minimale quand $ɛ \to 0^{+}$ . 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