Multiplicity via symmetry for the Minkowski mean curvature equation: Dirichlet and Neumann problems
Abstract
We consider the Minkowski mean curvature equation −div((∇ u)/(√(1−|∇ u|2))) =λu+μh(x,u) in Ω, ℬ u = 0 on ∂ Ω, where ℬ ∈ { 𝒟, 𝒩 } is a Dirichlet or Neumann boundary condition and Ω is a smooth bounded domain, required to be convex in the Neumann case. Moreover, λ, μ≥0 are parameters, and h(x,u) is odd with respect to u and positive for u > 0. Using minmax methods in nonsmooth critical point theory, we prove that λ>λ Bk yields at least k pairs of nontrivial solutions with negative energy for every μ≥0, where λk B are the corresponding Laplacian eigenvalues, indexed by λ B0=0. Moreover, if H(x,u):=∫0u h(x,t) dt=o(u2) as u→0, we prove that for λ∈[λ Bk,λ Bk+1) and every j≥max{1,k}, sufficiently large μ yields at least 2j−k distinct pairs of solutions: j with negative energy and j−k with positive energy.
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BibTeX
@misc{boscaggin2026multiplicity,
title = {{Multiplicity via symmetry for the Minkowski mean curvature equation: Dirichlet and Neumann problems}},
author = {Alberto Boscaggin and Ricardo Ziegele},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.07963},
eprint = {2611.07963},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.07963},
}
RIS
TY - UNPB
TI - Multiplicity via symmetry for the Minkowski mean curvature equation: Dirichlet and Neumann problems
AU - Alberto Boscaggin
AU - Ricardo Ziegele
PY - 2026
DA - 2026/11/05
PB - arXiv
DO - 10.48550/arxiv.2611.07963
UR - https://arxiv.org/abs/2611.07963
AB - We consider the Minkowski mean curvature equation −div((∇ u)/(√(1−|∇ u|^2))) =λu+μh(x,u) in Ω, ℬ u = 0 on ∂ Ω, where ℬ ∈ { 𝒟, 𝒩 } is a Dirichlet or Neumann boundary condition and Ω is a smooth bounded domain, required to be convex in the Neumann case. Moreover, λ, μ≥0 are parameters, and h(x,u) is odd with respect to u and positive for u > 0. Using minmax methods in nonsmooth critical point theory, we prove that λ>λ^ B_k yields at least k pairs of nontrivial solutions with negative energy for every μ≥0, where λ_k^ B are the corresponding Laplacian eigenvalues, indexed by λ^ B_0=0. Moreover, if H(x,u):=∫_0^u h(x,t) dt=o(u^2) as u→0, we prove that for λ∈[λ^ B_k,λ^ B_k+1) and every j≥max{1,k}, sufficiently large μ yields at least 2j−k distinct pairs of solutions: j with negative energy and j−k with positive energy.
ER -