Perturbations of Paneitz-Branson operators on $S^n$

Pierpaolo Esposito

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Some existence results for the scalar curvature problem via Morse theory

Andrea Malchiodi (1999)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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We prove existence of positive solutions for the equation $- △_{g_{0}} u + u = (1 + ϵ K (x)) u^{2^{*} - 1}$ on $S^{n}$ , arising in the prescribed scalar curvature problem. is the Laplace-Beltrami operator on $S^{n}$ , $2^{*}$ is the critical Sobolev exponent, and $ϵ$ is a small parameter. The problem can be reduced to a finite dimensional study which is performed with Morse theory.