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The paper deals with a class of Schrödinger-Poisson systems, where the coupling term and the other coefficients do not have any symmetry property. Moreover, the setting we consider does not allow the existence of ground state solutions. Under suitable assumptions on the decay rate of the coefficients, we prove existence of a bound state, finite energy solution.

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The paper deals with a semilinear elliptic Dirichlet problem with jumping\ud nonlinearity and, using variational methods, it shows that the number\ud of solutions tends to infinity as the number of jumped eigenvalues tends\ud to infinity.\ud In order to prove this fact, for every positive integer k it is proved\ud that, when a parameter is large enough, there exists a solution which\ud presents k interior peaks.\ud The asymptotic behaviour and the profile of this\ud solution, as the parameter tends to infinity, are also described

Using variational methods we prove some results about existence and multiplicity of positive bound states of to the following Schrödinger-Poisson system:We remark that (SP ) exhibits a "double" lack of compactness because of the unboundedness of R 3 and the critical growth of the nonlinear term and that in our assumptions ground state solutions of (SP ) do not exist. MSC: 35J20, 35J60