On conformal scalar curvature equations in \({\mathbb{R}}^ n\)
type: Item
instance of: scholarly article
title: On conformal scalar curvature equations in \({\mathbb{R}}^ n\)
zbMATH Open document ID: 0674.53048
DOI: 10.1215/S0012-7094-88-05740-7
author: Yi Li
author: Wei-Ming Ni
published in: Duke Mathematical Journal
publication date: 1988-01-01T00:00:00Z
review text: Cet article concerne l'étude du comportement asymptotique des solutions positives sur \({\mathbb{R}}^ n\) (n\(\geq 3)\) de l'équation (E): \(\Delta\) u\(+Ku^ p=0\) avec \(p>1\). Dans le cas général les AA., par des majorations du potentiel newtonien et l'utilisation de l'identité de Pohozaev, montrent que (E) ne possède pas de solution u bornée vérifiant \(\liminf_{x\to \infty}u(x)=0,\) si \(K\in C^ 1({\mathbb{R}}^ n)\), \(K(x)=O(| x|^{\ell})\) \((\ell <-2)\) à l'infini et si la fonction \([n-(n-2)(p+1)/2]K(x)+x\cdot \nabla K(x)\) ne change pas de signe sur \({\mathbb{R}}^ n\). Lorsque la fonction K est radiale, positive et \(\sim r^{\ell}\) (\(\ell \geq -2)\) à l'infini, les AA. s'intéressent aux solutions u radiales de (E) et décrivent leurs deux comportements asymptotiques possibles: (i) si \(\ell >-2\) alors \(u\sim r^{2-n}\) ou \(u\sim r^{-m}\) à l'infini avec \(m=(\ell +2)/(p-1)\) si l'on suppose n- 2\(\geq 2m\) et \((r^{-\ell}K(r))_ r\leq 0\) pour r grand (ii) si \(l=-2\) alors \(u\sim r^{2-n}\) ou \(u\sim (\log r)^{-1/(p-1)}\) à l'infini en faisant une hypothèse sur \(\liminf_{r\to \infty}r^ 2K(r).\) En ajoutant l'hypothèse \(K(r)=O(r^{\sigma})\) en 0 avec \(\sigma\geq 0\) et \(n+2+2\sigma >p(n-2)>n+2+2\ell,\) si max K\(>0\), les AA. montrent par un procède de minimisation que (E) possède une solution radiale \(u\geq 0\) telle que \(u\sim r^{2-n}\) à l'infini. Utilisant un résultat antérieur, ils exhibent, modulo quelques hypothèses supplementaires sur K, une infinité de solutions radiales de (E) ayant le second comportement asymptotique.
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 58J60
zbMATH DE Number: 4103916
zbMATH Keywords: conformal metrics
zbMATH Keywords: radial solutions
zbMATH Keywords: scalar curvature
zbMATH Keywords: asymptotic behaviour
zbMATH Keywords: Laplacian
reviewed by: Thierry Aubin
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4103916