Positive solutions for an indefinite-weight Minkowski mean curvature Neumann problem: multiplicity and asymptotic behaviour

Abstract

We study the Neumann boundary value problem $$ -\operatorname{div}\left(\frac{\nabla u}{\sqrt{1-|\nabla u|^2}}\right) = \lambda a(x)g(u) \quad \text{in } \Omega, \qquad \frac{\nabla u}{\sqrt{1-|\nabla u|^2}}\cdot \mathbf n = 0 \quad \text{on } \partial\Omega, $$ where $\Omega \subset \mathbb R^N$ is a bounded convex domain, $a\in L^\infty(\Omega)$ is an indefinite weight with $\int_\Omega a(x) \, dx<0$, and $g$ is a nonlinearity. Under suitable assumptions on $g$, we prove the existence of two positive solutions for sufficiently large $\lambda>0$, using Szulkin's theory for nonsmooth functionals: a global minimizer $u_\lambda^{(l)}$ with negative energy, and a mountain-pass critical point $u_\lambda^{(s)}$ with positive energy. Furthermore, we study the asymptotic behaviour as $\lambda\to+\infty$ of both families of solutions. For the ground states we prove that, up to a subsequence, $u_\lambda^{(l)}\to u_\infty$ uniformly, where $u_\infty$ satisfies $\|\nabla u_\infty\|_{L^\infty(\Omega)}=1$ and is characterized by being a maximizer of $\int_\Omega a(x) G(u) \, dx$ and a minimizer of the kinetic energy among all such maximizers. In the model case $g(u)=|u|^{p-2}u$, $p>2$, we also show that the mountain-pass energy level satisfies the upper bound $c_\lambda=O(\lambda^{-2/(p-2)})$, and that $u_\lambda^{(s)}\to0$ in $C(\overline\Omega)$.