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.
Type: Article
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)

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