Electron. J. Diff. Eqns., Vol. 2007(2007), No. 177, pp. 1-9.

Existence of positive solutions for p(x)-Laplacian problems

Ghasem A. Afrouzi, Horieh Ghorbani

Abstract:
We consider the system of differential equations
$$\displaylines{ -\Delta_{p(x)} u=\lambda [g(x)a(u) + f(v)] \quad\hbox{in }\Omega\cr -\Delta_{q(x)} v=\lambda [g(x)b(v) + h(u)] \quad\hbox{in }\Omega\cr u=v= 0 \quad\hbox{on } \partial \Omega }$$
where $p(x) \in C^1(\mathbb{R}^N)$ is a radial symmetric function such that $\sup|\nabla p(x)| < \infty$, $1 < \inf p(x) \leq \sup p(x) < \infty$, and where $-\Delta_{p(x)} u = -\hbox{div}|\nabla u|^{p(x)-2}\nabla u$ which is called the $p(x)$-Laplacian. We discuss the existence of positive solution via sub-super-solutions without assuming sign conditions on $f(0),h(0)$.

Submitted July 18, 2007. Published December 17, 2007.
Math Subject Classifications: 35J60, 35B30, 35B40
Key Words: Positive radial solutions; p(x)-Laplacian problems; boundary value problems.

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Ghasem A. Afrouzi
Department of Mathematics
Faculty of Basic Sciences
Mazandaran University, Babolsar, Iran
email: afrouzi@umz.ac.ir
Horieh Ghorbani
Department of Mathematics
Faculty of Basic Sciences
Mazandaran University, Babolsar, Iran
email: seyed86@yahoo.com

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