Electron. J. Diff. Equ., Vol. 2011 (2011), No. 12, pp. 1-9.

Existence of positive solutions for some nonlinear elliptic systems on the half space

Noureddine Zeddini

Abstract:
We prove some existence of positive solutions to the semilinear elliptic system
$$\displaylines{ \Delta u =\lambda p(x)g(v)\cr \Delta v =\mu q(x)f(u) }$$
in the half space ${\mathbb{R}}^n_+$, $n\geq 2$, subject to some Dirichlet conditions, where $\lambda$ and $\mu$ are nonnegative parameters. The functions $f, g$ are nonnegative continuous monotone on $(0,\infty)$ and the potentials $p, q$ are nonnegative and satisfy some hypotheses related to the Kato class $K^\infty({\mathbb{R}}^n_+)$.

Submitted March 16, 2010. Published January 21, 2011.
Math Subject Classifications: 35J55, 35J60, 35J65.
Key Words: Green function; Kato class; elliptic systems; positive solutions.

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Noureddine Zeddini
Department of Mathematics, College of Sciences and Arts
King Abdulaziz University, P.O. Box 344. Rabigh 21911, Saudi Arabia.
Département de Mathématiques, Faculté des Sciences de Tunis
Campus Universitaire, 2092 Tunis, Tunisia
email: noureddine.zeddini@ipein.rnu.tn

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