A saddle point type solution for a system of operator equations

摘要

Let Ω?Rn, n>1 and let p,q≥2. We consider the system of nonlinear Dirichlet problems?????????(Au)(x)?(Bv)(x)u(x)v(x)=N′u(x,u(x),v(x)),=N′v(x,u(x),v(x)),=0,=0,x∈Ω,x∈Ω,x∈?Ω,x∈?Ω,where N:R×R→R is C1 and is partially convex-concave and A:W1,p0(Ω)→(W1,p0(Ω))?, B:W1,p0(Ω)→(W1,p0(Ω))? are monotone and potential operators. The solvability of this system is reached via the Ky–Fan minimax theorem.