EXISTENCE OF SOLUTIONS FOR SYSTEMS OF MINKOWSKI-CURVATURE NEUMANN PROBLEMS

Abstract

We show some existence results for the system of nonlocal Neumann problems with the Minkowski-curvature operator

$${\left(r^{N-1}\frac{u^{\prime }}{\sqrt{1-{u^{\prime }}^2}}\right)}^{\prime }=r^{N-1}f(r,u,u^{\prime }),r\in (0,1),u^{\prime }(0)=0,u^{\prime }(1)={\displaystyle {\int }_0^1{u}^{\prime }(s)dg(s)},$$

where $N\ge 1$ is an integer, $f:[0,1]\times {ℝ}^k\times I^k\to {ℝ}^k$ is continuous and bounded, $I≔(-1,1)$, and $g:[0,1]\to {ℝ}^k$ is a function of bounded variation. The proof is based on topological-degree arguments and extends to a larger class of nonlinearities.