Anisotropic elliptic $\overrightarrow{p}(\cdot)-$Laplacian systems

Abstract

Within the framework of this paper, we aim to prove the existence of distributional solutions in the space $\mathring{W}^{1,\overrightarrow{p}(\cdot)}(\Omega,\mathbb{R}^m)$ for a new kind of nonlinear elliptic $\overrightarrow{p}(\cdot)-$anisotropic Laplace systems, such that its right-hand side is a nonlinearity connecting the solution $u$ and given functions $\psi_i\in L^{p_i(\cdot)}(\Omega,\mathbb{R}^m),\,i=1,\ldots,N.$