Electronic Journal of Differential Equations, Vol. 2012 (2012), No. 58, pp. 1–9. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu EXISTENCE OF BOUNDED POSITIVE SOLUTIONS OF A NONLINEAR DIFFERENTIAL SYSTEM SABRINE GONTARA Abstract. In this article, we study the existence and nonexistence of solutions for the system 1 A (Au ′ ) ′ = pu α v s on (0, ∞), 1 B (Bu ′ ) ′ = qu r v β on (0, ∞), Au ′ (0) = 0, u(∞)= a> 0, Bv ′ (0) = 0, v(∞)= b> 0, where α, β ≥ 1, s, r ≥ 0, p, q are two nonnegative functions on (0, ∞) and A, B satisfy appropriate conditions. Using potential theory tools, we show the existence of a positive continuous solution. This allows us to prove the existence of entire positive radial solutions for some elliptic systems. 1. Introduction Existence and nonexistence of solutions of the elliptic system Δu = p(|x|)f (v),x ∈ R n Δv = q(|x|)g(u),x ∈ R n (1.1) have been intensively studied in the previous years; see for example [2, 3, 4, 5, 6, 9, 10] and the references therein. Lair and Wood [6] considered the existence of entire positive radial solutions to the system (1.1) when f (v)= v s and g(u)= u r . More precisely, for the sublinear case where r,s ∈ (0, 1), they proved that if p and q satisfy the decay conditions  ∞ 0 tp(t)dt< ∞,  ∞ 0 tq(t)dt< ∞ (1.2) then (1.1) has bounded solutions, and if  ∞ 0 tp(t)dt = ∞,  ∞ 0 tq(t)dt = ∞ 2000 Mathematics Subject Classification. 35J56, 31B10, 34B16, 34B27. Key words and phrases. Nonlinear equation; Green’s function; asymptotic behavior; singular operator; positive solution. c 2012 Texas State University - San Marcos. Submitted January 20, 2012. Published April 10, 2012. 1