## Sur un problème variationnel non compact: L’effet de petits trous dans le domaine. (On a variational problem with lack of compactness: The effect of small holes in the domain).(French)Zbl 0686.35047

We are interested in the problem: $$-\Delta u=u^{(N+2)/(N-2)}$$, $$u>0$$ on $$\Omega_ d$$; $$u=0$$ on $$\partial \Omega_ d$$, where $$\Omega_ d$$ is a smooth and bounded domain in $${\mathbb{R}}^ N$$, $$N\geq 3$$, deleted from p discs of radius d. We show that, for d small enough, the equation has at least p solutions, each one concentrating around a hole as d tends to zero. Moreover, if the matrix $$(a_{ij})_{1\leq i,j\leq p}$$ defined by: $$a_{ii}=H(X_ i,X_ i)$$, $$a_{ij}=-G(X_ i,X_ j)$$ for $$i\neq j$$- where the $$X_ i's$$ are the centers of the discs, G is the Green’s function of the Laplace operator on $$\Omega$$ and H is its regular part - is positive definite, we show that for d small enough the equation has at least $$2^ p-1$$ solutions, which concentrate around the holes as d tends to zero.
Reviewer: O.Rey

### MSC:

 35J65 Nonlinear boundary value problems for linear elliptic equations 35B99 Qualitative properties of solutions to partial differential equations