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

Khafagy, Salah A. ; Serag, Hassan M.

Electronic journal of differential equations, Tome 2009 (2009)

Résumé

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\neq 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 $\Delta_{p}u\equiv \hbox{div} [|\nabla u|^{p-2}\nabla u]$ with $p>1, p\neq 2; \alpha _i\geq 0; f_i$ are given functions; and the coefficients $a_{ij}(x) ( 1\leq i,j\leq n)$ 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.

Détail

Classification : 74H20, 35J65
Keywords: existence of weak solution, nonlinear system, p-Laplacian