A priori estimates and existence of solutions to a system of nonlinear elliptic equations

We consider the following nonlinear elliptic equations−Δu+u+λϕ(x)|u|r−2u=|u|p−1u,x∈Ω,−Δϕ(x)=|u|q,x∈Ω,ϕ(x)=u(x)=0 for x∈∂Ω,(P)where p,q,r>1, λ is a parameter and Ω⊂R3 is a bounded domain. For q=r=2, the equations reduce to the Schrödinger-Poisson equations. Without the need of imposing constraint that q must be equal to r, we establish a priori estimates, the nonexistence and existence of solutions for problem (P). Our results extend previous work for the case q=r to more general case.