Su alcune successioni di soluzioni positive di problemi ellittici con esponente critico

Passaseo, Donato

Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni, Série 9, Tome 3 (1992) no. 1, pp. 15-21.

Riassunto (VO)

— Si presentano alcuni risultati di esistenza e molteplicità di soluzioni positive per l'equazione \( \Delta u + u^{2^{*}-1} = 0 \) in \( H_{0}^{(1,2)} (\Omega) \), dove \( \Omega \) è un aperto limitato di \( \mathbb{R}^{n} \) con \( n \ge 3 \) e \( 2^{*} = 2n / (n — 2) \). Si mostra che opportune perturbazioni di \( \Omega \) comportano l'esistenza di soluzioni positive, che convergono a zero quando la capacità delle perturbazioni tende a zero. In particolare, si ottengono risultati di esistenza e molteplicità di soluzioni positive in alcuni aperti limitati e contrattili, non necessariamente simmetrici.

Abstract

We present some existence and multiplicity results of positive solutions of the equation \( \Delta u + u^{2^{*}-1} = 0 \) in \( H_{0}^{(1,2)} (\Omega) \), where \( \Omega \) is a bounded domain of \( \mathbb{R}^{n} \) with \( n \ge 3 \) and \( 2^{*} = 2n / (n — 2) \). We show that suitable perturbations of \( \Omega \) give the existence of positive solutions which converge to zero as the capacity of the perturbations goes to zero. In particular, we obtain existence and multiplicity results of positive solutions in some bounded contractible domains, without any simmetry assumption.

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