Positive solutions for an indefinite-weight Minkowski mean curvature Neumann problem: multiplicity and asymptotic behaviour
Year: 2026
Type: preprint
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 $Ω\subset \mathbb R^N$ is a bounded convex domain, $a\in L^\infty(Ω)$ is an indefinite weight with $\int_Ωa(x) \, dx< 0$, and $g$ is a nonlinearity. Under suitable assumptions on $g$, we prove the existence of two positive solutions for sufficiently large $λ>0$, using Szulkin's theory for nonsmooth functionals: a global minimizer $u_λ^{(l)}$ with negative energy, and a mountain-pass critical point $u_λ^{(s)}$ with positive energy. Furthermore, we study the asymptotic behaviour as $λ\to+\infty$ of both families of solutions. For the ground states we prove that, up to a subsequence, $u_λ^{(l)}\to u_\infty$ uniformly, where $u_\infty$ satisfies $\|\nabla u_\infty\|_{L^\infty(Ω)}=1$ and is characterized by being a maximizer of $\int_Ωa(x) G(u) \, dx$ 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 $c_λ=O(λ^{-2/(p-2)})$, and that $u_λ^{(s)}\to0$ in $C(\overlineΩ)$.
Source: arXiv (Cornell University)
Author: Ziegele, Ricardo
Institution: University of Turin
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7212173016