Pohozaev type obstructions and solutions of bounded energy for quasilinear elliptic equations with critical Sobolev growth. The conformally flat case
type: Item
instance of: scholarly article
title: Pohozaev type obstructions and solutions of bounded energy for quasilinear elliptic equations with critical Sobolev growth. The conformally flat case
zbMATH Open document ID: 1066.35032
DOI: 10.1016/S0362-546X(01)00813-6
author: Olivier Druet
author: Emmanuel Hebey
author: Michel Vaugon
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2002-09-17T00:00:00Z
review text: Let \((M,g)\) be a smooth, compact, Riemannian manifold of dimension \(n\geq 3\), and consider the equation \[ \Delta_g u + \alpha u = u^{2^*-1}, \qquad u>0 \quad\text{in}\quad M, \leqno(*)_\alpha \] where \(\Delta_g u=-\text{ div}_g(\nabla u)\) and \(2^*=2n/(n-2)\) is the critical Sobolev exponent. If \(\Omega\) is a smooth, bounded, star-shaped domain in \({\mathbb R}^n\) with the Euclidean metric, and \(u\) is required to vanish on \(\partial\Omega\), the well known Pohozaev identity implies that there is no positive solution of \((*)_\alpha\) with \(M=\Omega\) if \(\alpha\geq 0\). Here the authors are interested in proving some generalization of this result in the Riemannian setting. Observing first that there is no solution of \((*)_\alpha\) if \(\alpha=0\), and that \(u_\alpha=\alpha^{(n-2)/4}\) is a solution for \(\alpha>0\), one possible generalization would be that there exists an \(\alpha_0\) such that \((*)_\alpha\) has no nonconstant solution if \(\alpha\geq \alpha_0\). However, this turns out to be false: the authors exhibit examples of conformally flat manifolds \((M,g)\) for which there are nonconstant solutions of \((*)_\alpha\) with \(\alpha\geq\alpha_0\) for arbitrarily large \(\alpha_0\). The main result of the paper is the following. Define \[ {\mathcal S}_\Lambda = \left\{ \alpha>0: \text{there exists a solution \(u\) of \((*)_\alpha\) with \(\left(\int_M u^{2^*}\right)^{1/{2^*}} \leq \Lambda\) } \right\}. \] Then, if \((M,g)\) is conformally flat, \({\mathcal S}_\Lambda\) is bounded for any \(\Lambda>0\). Moreover, \({\mathcal S}_\Lambda\) is closed in the set of positive real numbers if \(\Lambda\) is sufficiently large. The proof is by a blow up analysis. The main difficult is to deal with blowing up sequences that have multiple concentration points. This result was proved by \textit{E.~Hebey} and \textit{M.~Vaugon} [Duke Math. J. 79, 235--279 (1995; Zbl 0839.53030)] in the special case that \(\Lambda=K(n,2)^{-2/(2^*-2)}\) where \(K(n,2)\) is…
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 35B33
Mathematics Subject Classification ID: 58J05
zbMATH DE Number: 1801696
zbMATH Keywords: quasilinear elliptic equations
zbMATH Keywords: Riemannian manifolds
zbMATH Keywords: Pokhozhaev obstruction
zbMATH Keywords: critical Sobolev exponent
reviewed by: John Urbas
MaRDI profile type: Publication
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full work available at URL: s0362-546x(01)00813-6
OpenAlex ID: W4298193613
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description: scientific article; zbMATH DE number 1801696