Blow-Up Behaviors of Minimizers for the H˙ c-Critical Fourth-Order Nonlinear Schrödinger Equation with the Mixed Dispersions

• Yichun Mo ^[1] ; Abdoulaye Ali Youssouf ^[2] ; Binhua Feng ^[2]
  1. [1] Lanzhou Jiaotong University

     Lanzhou Jiaotong University

     China

     
  2. [2] Northwest Normal University

     Northwest Normal University

     China

     
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 24, Nº 2, 2025
• Idioma: inglés
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• Resumen

  • In this paper, we consider blow-up behaviors of constraint minimizers for the H˙ γc - critical fourth-order nonlinear Schrödinger equation with the mixed dispersions iψt − 2ψ + μψ + |ψ| pψ = 0.

    This equation arises in describing the propagation of intense laser beams in a bulk medium with Kerr nonlinearity. This paper seems to be the first time to present and study the minimizing problem:

    m1(c) := inf E(u), u ∈ H˙ γc ∩ H˙ 2 and uH˙ γc = c , where γc := N 2 − 4 p is the critical Sobolev exponent and E(u) = 1 2 u2 L2 + μ 2 ∇u2 L2 − 1 p + 2 up+2 L p+2 .

    Minimizers of this problem exist only if c < QH˙ γc , where Q is a solution of equation 2Q + (−)γc Q − |Q|pQ = 0. We then give a detailed description of blow-up behavior of minimizers as c QH˙ γc