Super-critical boundary bubbling in a semilinear Neumann problem

  • Manuel del Pino

    Departamento de Ingeniería Matemática and CMM, Universidad de Chile, Casilla 170 Correo 3, Santiago, Chile
  • Monica Musso

    Dipartimento di Matematica, Politecnico di Torino, Corso Duca degli Abruzzi, 24-10129 Torino, Italy
  • Angela Pistoia

    Dipartimento di Metodi e Modelli Matematici per le Scienze Applicate, Universitá di Roma a Sapienza, Via Scarpa 16, 00161 Roma, Italy

Abstract

In this paper we consider the following problem

{Δu+u=uN+2N2+ɛinΩ,u>0inΩ,uν=0onΩ,\left\{\begin{matrix} −\mathrm{\Delta }u + u = u^{\frac{N + 2}{N−2} + ɛ}\text{} & \text{in}\:\Omega \text{,} \\ u > 0\text{} & \text{in}\:\Omega \text{,} \\ \frac{\partial u}{\partial \nu } = 0\text{} & \text{on}\:\partial \Omega \text{,} \\ \end{matrix}\right.

where Ω is a smooth bounded domain in RN\mathbb{R}^{N} and N3N⩾3.

We prove the existence of a one-spike solution to (0.1) which concentrates around a topologically non trivial critical point of the mean curvature of the boundary with positive value. Under some symmetry assumption on Ω, namely if Ω is even with respect to N1N−1 variables and 0Ω0 \in \partial \Omega is a point with positive mean curvature, we prove existence of solutions to (0.1) which resemble the form of a super-position of spikes centered at 0.

Résumé

Dans cet article nous considérons le problème suivant :

{Δu+u=uN+2N2+ɛdansΩ,u>0dansΩ,uν=0surΩ,\left\{\begin{matrix} −\mathrm{\Delta }u + u = u^{\frac{N + 2}{N−2} + ɛ}\text{} & \text{dans}\:\Omega \text{,} \\ u > 0\text{} & \text{dans}\:\Omega \text{,} \\ \frac{\partial u}{\partial \nu } = 0\text{} & \text{sur}\:\partial \Omega \text{,} \\ \end{matrix}\right.

Ω est un domaine borné régulier dans RN\mathbb{R}^{N} et N3N⩾3. Nous prouvons l'existence d'une solution 1-transitoire au problème (0.2), qui se concentre autour d'un point critique topologiquement non trivial de la courbure moyenne, où celle-ci est strictement positive. Sous certaines hypothéses de symétrie sur Ω nous prouvons l'existence de solutions de (0.2) qui ressemblent à une superposition des transitoires centrées en un certain point du bord.

Cite this article

Manuel del Pino, Monica Musso, Angela Pistoia, Super-critical boundary bubbling in a semilinear Neumann problem. Ann. Inst. H. Poincaré Anal. Non Linéaire 22 (2005), no. 1, pp. 45–82

DOI 10.1016/J.ANIHPC.2004.05.001