Existence of nontrivial solution for a 4-sublinear Schrödinger–Poisson system

We consider a Schrödinger–Poisson system in R 3 with potential indefinite in sign and a general 4-sublinear nonlinearity. Its variational functional does not satisfy the Palais–Smale condition in general. We use a finite-dimensional approximation method to obtain a sequence of approximate solutions. Then we prove that these approximate solutions converge weakly to a nontrivial solution of this system.