## Multi-peak solutions for super-critical elliptic problems in domains with small holes.(English)Zbl 1014.35028

This paper is devoted to the construction of solutions of the problem $\begin{cases} -\Delta u=u^{{N+2 \over N-2}+\varepsilon} \quad &\text{in } \Omega,\\ u>0\quad &\text{in }\Omega,\\ u=0 \quad &\text{on }\partial\Omega, \end{cases}\tag{1}$ where $$\Omega$$ is a bounded domain with smooth boundary in $$\mathbb{R}^N$$, $$N\geq 3$$, and $$\varepsilon >0$$ is a small parameter. The goal of this paper is to raise the issue of solvability (1) in a domain exhibiting multiple holes. The main result of the authors states that in such a situation, multi-peak solutions exist, consisting of the glueing of double-spikes associated with each of the holes.

### MSC:

 35J60 Nonlinear elliptic equations

### Keywords:

multiple holes; multi-peak solutions; positive solution
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### References:

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