An Existence Theorem for Semi-linear Elliptic Systems
Henghui Zou · Advanced Nonlinear Studies · 2013
Abstract We study the following boundary value problem of semi-linear elliptic systems Δu + f(x, u, ∇u) =0 in Ω, u =0 on ∂Ω where Ω ⊂ 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..