Positive solutions for an indefinite-weight Minkowski mean curvature Neumann problem: multiplicity and asymptotic behaviour
Abstract
We study the Neumann boundary value problem −div((∇ u)/(√(1−|∇ u|2))) = λa(x)g(u) in Ω, (∇ u)/(√(1−|∇ u|2))· n = 0 on ∂Ω, where Ω⊂ ℝN is a bounded convex domain, a∈ L∞(Ω) is an indefinite weight with ∫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 uλ(l) with negative energy, and a mountain-pass critical point uλ(s) with positive energy. Furthermore, we study the asymptotic behaviour as λ→+∞ of both families of solutions. For the ground states we prove that, up to a subsequence, uλ(l)→ u∞ uniformly, where u∞ satisfies |∇ u∞|L∞(Ω)=1 and is characterized by being a maximizer of ∫Ωa(x) G(u) dx and a minimizer of the kinetic energy among all such maximizers. In the model case g(u)=|u|p−2u, p>2, we also show that the mountain-pass energy level satisfies the upper bound cλ=O(λ−2/(p−2)), and that uλ(s)→0 in C(Ω̅).
Related
- Existence and multiplicity of solutions for a class of new nonlocal problem with indefinite potential
- Multiplicity via symmetry for the Minkowski mean curvature equation: Dirichlet and Neumann problems
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BibTeX
@misc{ziegele2026positive,
title = {{Positive solutions for an indefinite-weight Minkowski mean curvature Neumann problem: multiplicity and asymptotic behaviour}},
author = {Ricardo Ziegele},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.09894},
eprint = {2609.09894},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.09894},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Positive solutions for an indefinite-weight Minkowski mean curvature Neumann problem: multiplicity and asymptotic behaviour
AU - Ricardo Ziegele
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2609.09894
UR - https://arxiv.org/abs/2609.09894
AB - We study the Neumann boundary value problem −div((∇ u)/(√(1−|∇ u|^2))) = λa(x)g(u) in Ω, (∇ u)/(√(1−|∇ u|^2))· n = 0 on ∂Ω, where Ω⊂ ℝ^N is a bounded convex domain, a∈ L^∞(Ω) is an indefinite weight with ∫_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→+∞ of both families of solutions. For the ground states we prove that, up to a subsequence, u_λ^(l)→ u_∞ uniformly, where u_∞ satisfies |∇ u_∞|_L^∞(Ω)=1 and is characterized by being a maximizer of ∫_Ωa(x) G(u) dx and a minimizer of the kinetic energy among all such maximizers. In the model case g(u)=|u|^p−2u, 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)→0 in C(Ω̅).
ER -