Positive Solutions for an Indefinite-Weight Minkowski Mean Curvature Neumann Problem: Multiplicity and Asymptotic Behaviour

Abstract

We study the Neumann boundary value problem $$-\mathrm{div}\left(\frac{\nabla u}{\sqrt{1-|\nabla u{|}^2}}\right)=\lambda a(x)g(u)\text{in }\Omega ,\frac{\nabla u}{\sqrt{1-|\nabla u{|}^2}}\cdot \mathbf{n}=0\text{on }\partial \Omega ,$$ where $\Omega \subset {\mathbb{R}}^N$ is a bounded convex domain, $a\in L^{\infty }(\Omega )$ is an indefinite weight with ${\int }_{\Omega }a<0$, 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 of both solutions as $\lambda \to +\infty$ in the model case $g(u)=|u{|}^{p-2}u$. We show that the mountain-pass energy level decays at the explicit rate $c_{\lambda }=O({\lambda }^{-2/(p-2)})$, and that $u_{\lambda }^{(s)}\to 0$ in $C(\overline{\Omega })$; moreover, we prove that $u_{\lambda }^{(l)}\to u_{\infty }$ uniformly, where $u_{\infty }$ solves a constrained maximization problem. The limiting profile $u_{\infty }$ saturates the geometric constraint, $\Vert \nabla u_{\infty }{\Vert }_{L^{\infty }(\Omega )}=1$, and on every connected open set where the gradient constraint is inactive and $a\ne 0$, the function $u_{\infty }$ is constant.