Electron. J. Diff. Equ., Vol. 2012 (2012), No. 239, pp. 1-10.

Positive solutions for nonlinear elliptic systems

Adel Ben Dekhil

Abstract:
In this article, we study the existence of positive solutions for the system
$$\displaylines{ \Delta u=H(x,u,v),\cr \Delta v=K(x,u,v),\hbox{in }\mathbb{R}^n\; (n\geq 3), }$$
where $H,K: \mathbb{R}^n\times[0,\infty)\times[0,\infty)\to[0,\infty)$ are continuous functions satisfying $H(x,u,v)\leq p_1(|x|)F(u+v)$ and $ K(x,u,v)\leq q_1(|x|)G(u+v)$. In terms of the growth of the variable potential functions $p_1,q_1$ and the nonlinearities F and G, we establish some sufficient conditions for the existence of positive continuous solutions for this system and we discuss whether these solutions are bounded or blow up at infinity.

Submitted October 14, 2012. Published December 28, 2012.
Math Subject Classifications: 35B08, 35B09, 35J47.
Key Words: Semilinear elliptic systems; positive large solution; positive bounded solution.

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Adel Ben Dekhil
Département de Mathématiques
Faculté des Sciences de Tunis
Université de Tunis El Manar
Campus Universitaire, 2092 Tunis, Tunisia
email: Adel.Bendekhil@ipein.rnu.tn

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