Please use this identifier to cite or link to this item: http://hdl.handle.net/10077/8311
Title: Infinitely many radial solutions of a mean curvature equation in Lorentz-Minkowski space
Authors: Bonheure, Denis
De Coster, Colette
Derlet, Ann
Keywords: Mean curvature equation in the Lorentz-Minkowski spaceLusternik- Schnirelmann categorymultiplicitysuper critical exponent
Issue Date: 2012
Publisher: EUT Edizioni Università di Trieste
Source: Denis Bonheure, Colette De Coster, Ann Derlet, "Infinitely many radial solutions of a mean curvature equation in Lorentz-Minkowski space", in: Rendiconti dell’Istituto di Matematica dell’Università di Trieste. An International Journal of Mathematics, 44 (2012), pp. 259–284.
Series/Report no.: Rendiconti dell’Istituto di Matematica dell’Università di Trieste. An International Journal of Mathematics
44 (2012)
Abstract: In this paper, we show that the quasilinear equation $$ -{\rm div}\left(\frac{\nabla u}{\sqrt{1-|\nabla u|^{2}}}\right) = |u|^{\alpha-2}u,\ \text{ in }\mathbb{R}^{N} $$ has a positive smooth radial solution at least for any $\alpha> 2^{\star}=2N/(N-2)$, $N\ge 3$. Our approach is based on the study of the optimizers for the best constant in the inequality $$ \int_{\mathbb{R}^N}(1-\sqrt{1-|\nabla u|^2}) \ge C \left( \int_{\mathbb{R}^{N}} |u|^\alpha\right)^{\frac{N}{\alpha+N}}, $$ which holds true in the unit ball of $W^{1,\infty}(\mathbb{R}^{{N}})\cap \mathcal D^{1;2}(\mathbb{R}^{N})$ if and only if $\alpha\ge 2^{\star}$. We also prove that the best constant is not achieved for $\alpha=2^{\star}$. As a byproduct, our arguments combined with Lusternik-Schnirelmann category theory allow to construct a sequence of radial solutions.
URI: http://hdl.handle.net/10077/8311
ISSN: 0049-4704
Appears in Collections:Rendiconti dell'Istituto di matematica dell'Università di Trieste: an International Journal of Mathematics vol.44 (2012)

Files in This Item:
File Description SizeFormat 
Bonheure_DeCoster_Derlet_RIMUT44.pdf310.56 kBAdobe PDFView/Open
Show full item record


CORE Recommender

Page view(s)

698
Last Week
0
Last month
0
checked on Oct 22, 2018

Download(s)

545
checked on Oct 22, 2018

Google ScholarTM

Check


Items in DSpace are protected by copyright, with all rights reserved, unless otherwise indicated.