Electron. J. Diff. Eqns., Vol. 2009(2009), No. 81, pp. 1-10.

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

Salah A. Khafagy, Hassan M. Serag

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 $\Delta_{p}u\equiv \hbox{div} [|\nabla u|^{p-2}\nabla u]$ 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.

Submitted December 9, 2008. Published July 10, 2009.
Math Subject Classifications: 74H20, 35J65.
Key Words: Existence of weak solution; nonlinear system, p-Laplacian.

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Salah A. Khafagy
Mathematics Department, Faculty of Science
Al-Azhar University, Nasr City 11884, Cairo, Egypt
email: el_gharieb@hotmail.com
Hassan M. Serag
Mathematics Department, Faculty of Science
Al-Azhar University, Nasr City 11884, Cairo, Egypt
email: serraghm@yahoo.com

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