Strongly coupled elliptic equations related to mean-field games systems

In this paper, we study existence of solutions for the following elliptic problem, related to mean-field games systems:{−div(M(x)∇ζ)+ζ−div(ζA(x)∇u)=fin Ω,−div(M(x)∇u)+u+θA(x)∇u⋅∇u=ζpin Ω,ζ=0=uon ∂Ω,where p>0, 0<θ<1, and f≥0 is a function in some Lebesgue space.