Blow-Up Behaviors of Minimizers for the Hγc-Critical Fourth-Order Nonlinear Schrodinger Equation with the Mixed Dispersions

In this paper, we consider blow-up behaviors of constraint minimizers for the H-gamma c-critical fourth-order nonlinear Schrodinger equation with the mixed dispersions i psi(t)-Delta(2)psi+mu Delta psi+|psi|(p)psi=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: m(1)(c):=inf{E(u), u is an element of H-gamma c boolean AND H-2 and & Vert;u & Vert;(H gamma c)=c}, where gamma c:=N/2-4/p is the critical Sobolev exponent and E(u)=1/2 & Vert;Delta u & Vert;(2)(L2)+mu/2 & Vert;del u & Vert;(2)(L2)-1/p+2 & Vert;u & Vert;(p+2)(Lp+2). Minimizers of this problem exist only if c<& Vert;Q & Vert;(H gamma c), where Q is a solution of equation Delta(2)Q+(-Delta)(gamma c)Q-|Q|(p)Q=0. We then give a detailed description of blow-up behavior of minimizers as c NE arrow & Vert;Q & Vert;(H gamma c).