Electronic Journal of Differential Equations,
Vol. 2007(2007), No. 177, pp. 1-9.
Title: Existence of positive solutions for p(x)-Laplacian problems
Authors: Ghasem A. Afrouzi (Mazandaran Univ., Babolsar, Iran)
Horieh Ghorbani (Mazandaran Univ., Babolsar, Iran)
Abstract:
We consider the system of differential equations
$$\displaylines{
-\Delta_{p(x)} u=\lambda [g(x)a(u) + f(v)] \quad\hbox{in }\Omega\cr
-\Delta_{q(x)} v=\lambda [g(x)b(v) + h(u)] \quad\hbox{in }\Omega\cr
u=v= 0 \quad\hox{on } \partial \Omega
}$$
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
$-\Delta_{p(x)} u = -{\rm div}|\nabla u|^{p(x)-2}\nabla u$
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.