Positive solutions of semilinear elliptic problems in unbounded domains with unbounded boundary

Abstract

This paper is concerned with the existence and multiplicity of positive solutions of the equation $-\mathrm{\Delta }u+u=u^{p-1}$, $2<p<2^{\ast }=\frac{2N}{N-2}$, with Dirichlet zero data, in an unbounded smooth domain $\Omega \subset {\mathbb{R}}^N$ having unbounded boundary. Under the assumptions:

$(h_1)\exists {\tau }_1,{\tau }_2,\dots ,{\tau }_k\in {\mathbb{R}}^{+}\setminus \{0\},1⩽k⩽N-2,\text{ such that }$

$(h_2)\exists R\in {\mathbb{R}}^{+}\setminus \{0\}\text{ such that }{\mathbb{R}}^N\setminus \Omega \subset \{(x_1,x_2,\dots ,x_N)\in {\mathbb{R}}^N\text{:}\sum _{j=k+1}^N{x}_j^2⩽R^2\}$

the existence of at least $k+1$ solutions is proved.

Résumé

Dans cet article on étudie l'existence et la multiplicité de solutions positives pour l'équation $-\mathrm{\Delta }u+u=u^{p-1}$, $2<p<2^{\ast }=\frac{2N}{N-2}$, avec la condition de Dirichlet $u=0$ sur $\partial \Omega$. Le domaine $\Omega \subset {\mathbb{R}}^N$ est non borné et $\partial \Omega$ est non borné aussi. En supposant que les conditions

$(h_1)\exists {\tau }_1,{\tau }_2,\dots ,{\tau }_k\in {\mathbb{R}}^{+}\setminus \{0\},1⩽k⩽N-2,\text{ tels que }$

$(h_2)\exists R\in {\mathbb{R}}^{+}\setminus \{0\}\text{ tel que }{\mathbb{R}}^N\setminus \Omega \subset \{(x_1,x_2,\dots ,x_N)\in {\mathbb{R}}^N\text{:}\sum _{j=k+1}^N{x}_j^2⩽R^2\}$

soient vérifiées, on démontre que le problème possède au moins $k+1$ solutions.