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

Abstract: We study the Neumann boundary value problem $$-\operatorname{div}\left(\frac{\nabla u}{\sqrt{1-|\nabla u|^2}}\right) = λa(x)g(u) \quad \text{in } Ω, \qquad \frac{\nabla u}{\sqrt{1-|\nabla u|^2}}\cdot \mathbf n = 0 \quad \text{on } \partialΩ,$$ where $Ω\subset \mathbb R<sup>N$ is a bounded convex domain, $a\in L<sup>\infty(Ω)$ is an indefinite weight with $\int_Ωa &lt; 0$, and $g$ is a nonlinearity. Under suitable assumptions on $g$, we prove the existence of two positive solutions for sufficiently large $λ&gt;0$, using Szulkin's theory for nonsmooth functionals: a global minimizer $u_λ<sup>{(l)}$ with negative energy, and a mountain-pass critical point $u_λ<sup>{(s)}$ with positive energy. Furthermore, we study the asymptotic behaviour of both solutions as $λ\to +\infty$ in the model case $g(u) = |u|^{p-2} u$. We show that the mountain-pass energy level decays at the explicit rate $c_λ= O(λ<sup>{-2/(p-2)})$, and that $u_λ^{(s)} \to 0$ in $C(\overlineΩ)$; moreover, we prove that $u_λ^{(l)} \to u_\infty$ uniformly, where $u_\infty$ solves a constrained maximization problem. The limiting profile $u_\infty$ saturates the geometric constraint, $|\nabla u_\infty|<em>{L<sup>\infty(Ω)}=1$, and on every connected open set where the gradient constraint is inactive and $a\neq 0$, the function $u</em>\infty$ is constant.