Let
Ω⊂Rn n>1 and let
p,q≥2. We consider the system of nonlinear Dirichlet problems
equation* brace(Au)(x)=Nu′(x,u(x),v(x)),x∈Ω,r-(Bv)(x)=Nv′(x,u(x),v(x)),x∈Ω,ru(x)=0,x∈∂Ω,rv(x)=0,x∈∂Ω,endequation* where
N:R×R→R is
C1 and is partially convex-concave and
A:W01,p(Ω)→(W01,p(Ω))*
B:W01,p(Ω)→(W01,p(Ω))* are monotone and potential operators. The solvability of this system is reached via the Ky–Fan minimax theorem.