An Existence Theorem for Semi-linear Elliptic Systems

We study the following boundary value problem of semi-linear elliptic systems Delta u + f(x, u, del u) = 0 in Omega, u = 0 on partial derivative Omega where Omega subset of R-n (n >= 2) is a connected and bounded smooth domain, and u and f are real vector-valued functions. Under appropriate conditions on the function f, we establish a general existence result for positive solutions. We continue our earlier works [15, 16, 17] and, in particular, remove a requirement of the cooperative structure on elliptic systems. As examples, we consider the stationary Klein-Gordon-Maxwell system and Maxwell-Schrodinger system, both lacking the cooperative structure..