On the existence of multiple positive solutions to some superlinear systems

where Ω is a smooth bounded domain in R, n 2, and gi(x, u, v) : Ω̄×R×R → R, i = 1, 2, are differentiable functions subject to further restrictions to be named below. We assume that gi(x, u, v), i = 1, 2, satisfy suitable conditions on a bounded rectangle so that a positive strict lower solution pair and a positive strict upper solution pair can be constructed. These assumptions suffice to prove the existence of at least one positive solution, provided the lower solution pair and upper solution pair are ordered componentwise. For example, in [4], Chhetri and Robinson prove the existence of a positive solution for a single equation analogous to (1.1) by constructing an ordered pair of lower and upper solutions. There, the nonlinearity is negative at the origin and satisfies additional conditions. (Further related references may be found therein.)