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

We consider the following nonlinear elliptic equations {-Delta u + u + lambda phi(x)vertical bar u vertical bar(r-2) u = vertical bar u vertical bar(p-1) u, x is an element of Omega, -Delta phi(x) =vertical bar u vertical bar(q), x is an element of Omega, (P) phi(x) =u(x) = 0, x is an element of partial derivative Omega, where p, q, r >1, lambda is a parameter and Omega subset of R-3 is a bounded domain. For q = r = 2, the equations reduce to the Schrodinger-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.