Infinitely many radial solutions of a mean curvature equation in Lorentz-Minkowski space
Résumé
In this paper, we show that the quasilinear equation - div(\frac{\nabla u}{\sqrt{1-|\nabla u|^{2}}}) = |u|^{\alpha-2}u, in R^N has a positive smooth radial solution at least for any α > 2 = 2N/(N − 2), N ≥ 3. Our approach is based on the study of the optimizers for the best constant in the inequality
\int_{R^N}(1-\sqrt{1-|\nabla u|^2}) \geq C ( \int_{R^N} |u|^\alpha )^{\frac{N}{\alpha+N}}, which holds true in the unit ball of W^{1,\infty}(R^N})\cap \mathcal D^{1;2}(R^N) if and only if α ≥ 2. We also prove that the best constant is not achieved for α = 2. As a byproduct, our arguments combined with Lusternik-Schnirelmann category theory allow to construct a sequence of radial solutions.
Origine : Fichiers produits par l'(les) auteur(s)