Multiplicity via symmetry for the Minkowski mean curvature equation: Dirichlet and Neumann problems

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.