Abstract
In this article, we study the existence and nonexistence of positive bounded solutions of the Dirichlet problem
-Delta u = lambda p(x)f(u, v), in R-+(n),
-Delta v = lambda q(x)g(u, v), in R-+(n),
u = v = 0 on partial derivative R-+(n),
lim(vertical bar x vertical bar ->infinity) u(x) = lim(vertical bar x vertical bar ->infinity) v(x) = 0,
where R-+(n) = {x = (x(1), x(2), ..., x(n)) is an element of R-n : x(n) > 0} (n >= 3) is the upper half-space and lambda is a positive parameter. The potential functions p, q are not necessarily bounded, they may change sign and the functions f, g : R-2 -> R are continuous. By applying the Leray-Schauder fixed point theorem, we establish the existence of positive solutions for lambda sufficiently small when f(0, 0) > 0 and g(0, 0) > 0. Some nonexistence results of positive bounded solutions are also given either if lambda is sufficiently small or if lambda is large enough.