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

Yichun Mo, Abdoulaye Ali Youssouf, Binhua Feng

• 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