Unité Mixte de Recherche du C.N.R.S. N°7534

Abstract : We consider the problem of finding positive solutions of $\Delta u + \lambda u + u^q = 0$ in a bounded, smooth domain $\Omega$ in $\RRN$, under zero Dirichlet boundary conditions. Here $q$ is a number close to the critical exponent $5$ and $0<\la<\la_1$. We analyze the role of Green's function of $\Delta +\la$ in the presence of solutions exhibiting single and multiple bubbling behavior at one point of the domain when either $q$ or $\la$ are regarded as parameters. As a special case of our results, we find that if $\la^*<\la <\la_1$, where $\la^*$ is the {\em Brezis-Nirenberg number,\/} i.e. the smallest value of $\la$ for which least energy solutions for $q=5$ exist, then this problem is solvable if $q>5$ and $q-5$ is sufficiently small.

 THE BREZIS-NIRENBERG PROBLEM NEAR CRITICALITY IN DIMENSION 3
 DEL PINO Manuel, DOLBEAULT Jean, MUSSO Monica
2004-5
20-01-2004

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