English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

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