Positive and nodal solutions for the Minkowski mean curvature equation: multiplicity and asymptotics

Abstract

We consider the Dirichlet problem for the mean curvature operator in Minkowski space, \[ -\operatorname{div}\left(\frac{\nabla u}{\sqrt{1-|\nabla u|^2}}\right) = \lambda u + \mu h(x,u) \quad \text{in } \Omega, \qquad u = 0 \quad \text{on } \partial\Omega, \] in a bounded domain $\Omega \subset \mathbb{R}^N$, where $\lambda, \mu$ are real parameters, and the nonlinearity $h$ is superlinear at $u = 0$. In particular, we study the combined effect of the parameters $\lambda,\,\mu$ on the multiplicity of solutions. In the general setting, following Szulkin's approach for nonsmooth functionals, we prove the existence, for $\lambda$ not belonging to the spectrum of the Dirichlet Laplacian and $\mu$ sufficiently large, of a global minimizing solution (with negative action level) and of a min-max solution (with positive action level). Moreover, we characterize the limiting profiles of these solutions as $\mu \to +\infty$. More precisely, when the global minimizer is positive, its limit profile is $\mathrm{dist}(\cdot,\partial\Omega)$, thus saturating, in the limit, the geometric constraint $|\nabla u|\le1$, while min-max solutions collapse uniformly to zero as $\mu\to+\infty$. A nonexistence criterion is also given for suitable values of $\lambda$ and $\mu$. Finally, when the domain $\Omega$ is a ball, using a shooting approach, we establish the existence of arbitrarily many nodal radial solutions for every $\lambda \ge 0$ and for $\mu$ sufficiently large.