Electronic Journal of Differential Equations,
Vol. 2009(2009), No. 81, pp. 1-10.
Title: Existence of weak solutions for nonlinear systems involving
several p-Laplacian operators
Authors: Salah A. Khafagy (Al-Azhar Univ., Nasr City, Cairo, Egypt)
Hassan M. Serag (Al-Azhar Univ., Nasr City, Cairo, Egypt)
Abstract:
In this article, we study nonlinear systems involving several p-Laplacian
operators with variable coefficients. We consider the system
$$
-\Delta _{p_i}u_i=a_{ii}(x)|u_i|^{p_i-2}u_i
-\sum_{j\neq i}^{n}a_{ij}(x)|u_i|^{\alpha _i}|u_j|^{\alpha_j}u_j+f_i(x),
$$
where $\Delta _p$ denotes the p-Laplacian defined by
$\Delta_{p}u\equiv \mathop{\rm div} [|\nabla u|^{p-2}\nabla u]$
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.
Submitted December 9, 2008. Published July 10, 2009.
Math Subject Classifications: 74H20, 35J65.
Key Words: Existence of weak solution; nonlinear system, p-Laplacian.