Existence of Solutions for Systems of $k$-Dimensional Minkowski-Curvature Problems with Neumann Conditions

Abstract:

      We prove the existence of solutions of the system for nonlocal Neumann problems with Minkowski-curvature operator $$(r^{N-1}\frac{u'}{\sqrt{1-u'^{2}}})'=r^{N-1}f(r, u),\ r\in(0, 1),\ \ u'(0)=0,\ u'(1)=\int_{0}^{1}u'(s)\d g(s),$$ where $k, N\geq1$ are integers, $f:[0, 1]\times\mathbb{R}^{k}\rightarrow\mathbb{R}^{k}$ is continuous and $g:[0, 1]\rightarrow\mathbb{R}^{k}$ is a function of bounded variation. Our proof is based on the perturbation method.