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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 space; Lusternik- Schnirelmann category; multiplicity; super 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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Bonheure_DeCoster_Derlet_RIMUT44.pdf | 310.56 kB | Adobe PDF | ![]() View/Open |
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