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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 3(  *  )                              i   ψ  t   +  Δ  ψ  −   (|   x   |−  1    *|   ψ   |    2    )   ψ  +|   ψ   |p  −  2    ψ  =  0    when    p  ∈   (10   3   ,  6   )    . To obtain such solutions, we look into critical points of the energy functional    F  (  u  )  =   1  2‖    ∇  u    ‖L  2   (   R  3   )   2   +   1  4∫    R  3 ∫    R  3 |    u  (  x  )    |    2|    u  (  y  )    |    2|    x  −  y    |d  x     d  y  −   1  p∫    R  3 |   u   |    p   d  x  ,    on the constraints given by    S  (  c  )  =   {    u  ∈   H  1(  R  )    3   :‖   u   ‖L  2    (    R  3    )    2   =  c  ,  c  >  0    }   .    For 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. Regarding 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.

Existence and instability of standing waves with prescribed norm for a class of Schrödinger–Poisson equations