Abstract. In this paper we study the existence and the instability of standing waves with prescribed L 2 -norm for a class of Schrödinger-Poisson-Slater equations in R 6). To obtain such solutions we look to critical points of the energy functionalon the constraints given byFor the values p ∈ ( 10 3 , 6) considered, the functional F is unbounded from below on S(c) and the existence of critical points is obtained by a mountain pass argument developed on S(c). We show that critical points exist provided that c > 0 is sufficiently small and that when c > 0 is not small a nonexistence result is expected. Concerning the dynamics we show for initial condition u 0 ∈ H 1 (R 3 ) of the associated Cauchy problem with u 0 2 2 = c that the mountain pass energy level γ(c) gives a threshold for global existence. Also the strong instability of standing waves at the mountain pass energy level is proved. Finally we draw a comparison between the Schrödinger-Poisson-Slater equation and the classical nonlinear Schrödinger equation.

In this paper we study the existence of minimizers forwhere c > 0 is a given parameter. In the range p ∈ [3, 10 3 ] we explicit a threshold value of c > 0 separating existence and non-existence of minimizers. We also derive a non-existence result of critical points of F (u) restricted to S(c) when c > 0 is sufficiently small. Finally, as a byproduct of our approaches, we extend some results of [9] where a constrained minimization problem, associated to a quasilinear equation, is considered.

In this paper we prove the existence of two solutions having a prescribed L 2 -norm for a quasi-linear Schrödinger equation. One of these solutions is a mountain pass solution relative to a constraint and the other one a minimum either local or global. To overcome the lack of differentiability of the associated functional, we rely on a perturbation method developed in [25].