Existence of weak solutions for nonlinear systems involving several p-Laplacian operators

Abstract

In this article, we study nonlinear systems involving several p-Laplacian operators with variable coefficients. We consider the system $$ -Delta _{p_i}u_i=a_{ii}(x)|u_i|^{p_i-2}u_i -sum_{j eq i}^{n}a_{ij}(x)|u_i|^{alpha _i}|u_j|^{alpha_j}u_j+f_i(x), $$ where $Delta _p$ denotes the p-Laplacian defined by $Delta_{p}uequiv mathop{ m div} [| abla u|^{p-2} abla u]$ with $p>1$, $p eq 2$; $alpha _igeq 0$; $f_i$ are given functions; and the coefficients $a_{ij}(x)$ ($1leq i,jleq n$) are bounded smooth positive functions. We prove the existence of weak solutions defined on bounded and unbounded domains using the theory of nonlinear monotone operators.

Keywords

• Existence of weak solution
• nonlinear system, p-Laplacian