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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Positive constrained minimizers for supercritical problems in the ball


Authors: Massimo Grossi and Benedetta Noris
Journal: Proc. Amer. Math. Soc. 140 (2012), 2141-2154
MSC (2010): Primary 35J25
Published electronically: October 24, 2011
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Abstract: We provide a sufficient condition for the existence of a positive solution to

$\displaystyle -\Delta u+V(\vert x\vert) u=u^p \quad \hbox { in } B_1, $

when $ p$ is large enough. Here $ B_1$ is the unit ball of $ \mathbb{R}^n$, $ n\ge 2$, and we deal with both Neumann and Dirichlet homogeneous boundary conditions. The solution turns out to be a constrained minimum of the associated energy functional. As an application we show that in case $ V(\vert x\vert)\geq 0$, $ V\not \equiv 0$ is smooth and $ p$ is sufficiently large, and the Neumann problem always admits a solution.

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Additional Information

Massimo Grossi
Affiliation: Dipartimento di Matematica, Università di Roma “La Sapienza”, P. le A. Moro 2, 00185 Roma, Italy
Email: grossi@mat.uniroma1.it

Benedetta Noris
Affiliation: Dipartimento di Matematica e Applicazioni, Università degli Studi di Milano-Bicocca, Via Bicocca degli Arcimboldi 8 - 20126 Milano, Italy
Email: benedettanoris@gmail.com

DOI: http://dx.doi.org/10.1090/S0002-9939-2011-11133-X
PII: S 0002-9939(2011)11133-X
Received by editor(s): July 11, 2010
Received by editor(s) in revised form: February 13, 2011
Published electronically: October 24, 2011
Communicated by: Matthew J. Gursky
Article copyright: © Copyright 2011 American Mathematical Society