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Infinitely many radial solutions of a mean curvature equation in Lorentz-Minkowski space
Bonheure, Denis
De Coster, Colette
Derlet, Ann
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.
Series
Rendiconti dell’Istituto di Matematica dell’Università di Trieste. An International Journal of Mathematics
44 (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.
Languages
en
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