Rey, Olivier 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 C. R. Acad. Sci., Paris, Sér. I 308, No. 12, 349-352 (1989). 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 Cited in 21 Documents MSC: 35J65 Nonlinear boundary value problems for linear elliptic equations 35B99 Qualitative properties of solutions to partial differential equations Keywords:variational problems with lack of compactness; limiting Sobolev exponent; Green’s function number of solutions; properties of solutions near small holes in the domain PDFBibTeX XMLCite \textit{O. Rey}, C. R. Acad. Sci., Paris, Sér. I 308, No. 12, 349--352 (1989; Zbl 0686.35047)