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


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