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

Abstract

We consider the Minkowski mean curvature equation $$ -\operatorname{div}\left(\frac{\nabla u}{\sqrt{1-|\nabla u|^2}}\right) =λu+μh(x,u) \quad \text{in } Ω, \quad \mathcal B u = 0 \quad \text{on } \partial Ω, $$ where $\mathcal B \in \{ \mathcal D, \mathcal N \}$ is a Dirichlet or Neumann boundary condition and $Ω$ is a smooth bounded domain, required to be convex in the Neumann case. Moreover, $λ, μ\geq0$ 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 $λ>λ^\mathcal B_k$ yields at least $k$ pairs of nontrivial solutions with negative energy for every $μ\geq0$, where $λ_k^\mathcal B$ are the corresponding Laplacian eigenvalues, indexed by $λ^\mathcal B_0=0$. Moreover, if $H(x,u):=\int_0^u h(x,t)\,dt=o(u^2)$ as $u\to0$, we prove that for $λ\in[λ^\mathcal B_k,λ^\mathcal B_{k+1})$ and every $j\geq\max\{1,k\}$, sufficiently large $μ$ yields at least $2j-k$ distinct pairs of solutions: $j$ with negative energy and $j-k$ with positive energy.