Topological methods for boundary value problems involving discrete vector $\phi$-Laplacians

Abstract

In this paper, using Brouwer degree arguments, we prove some existence results for nonlinear problems of the type $$ -\nabla[\phi(\Delta x_m)]=g_m(x_m,\Delta x_m) \quad (1\leq m\leq n-1), $$ submitted to Dirichlet, Neumann or periodic boundary conditions, where $\phi(x)=|x|^{p-2}x$ $(p> 1)$ or $\phi(x)={x}/{\sqrt{1-|x|^2}}$ and $g_m\colon \mathbb{R}^N\to\mathbb{R}^N$ $(1\leq m\leq n-1)$ are continuous nonlinearities satisfying some additional assumptions.