Infinitely many radial solutions of a mean curvature equation in Lorentz-Minkowski space

Abstract

In this paper, we show that the quasilinear equation
\n$$
\n-{\\rm div}\\left(\\frac{\\nabla u}{\\sqrt{1-|\\nabla u|^{2}}}\\right) = |u|^{\\alpha-2}u,\\ \\text{ in }\\mathbb{R}^{N}
\n$$
\nhas 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
\n$$
\n\\int_{\\mathbb{R}^N}(1-\\sqrt{1-|\\nabla u|^2}) \\ge C \\left( \\int_{\\mathbb{R}^{N}} |u|^\\alpha\\right)^{\\frac{N}{\\alpha+N}},
\n$$
\nwhich 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.