JournalsrmiVol. 32, No. 3pp. 795–833

Minimal mass blow up solutions for a double power nonlinear Schrödinger equation

  • Stefan Le Coz

    Université Paul Sabatier, Toulouse, France
  • Yvan Martel

    École Polytechnique, Palaiseau, France
  • Pierre Raphaël

    Université de Nice Sophia Antipolis, France
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Abstract

We consider a nonlinear Schrödinger equation with double power nonlinearity

itu+Δu+u4/du+ϵup1u=0,ϵ{1,0,1},1<p<1+4di\partial_tu +\Delta u+|u|^{4/d}u+\epsilon |u|^{p-1}u=0, \quad \epsilon\in\{-1,0,1\}, \quad 1 < p <1 + \frac 4d

in Rd\mathbb R^d (d=1,2,3d=1,2,3). Classical variational arguments ensure that H1(Rd)H^1(\mathbb R^d) data with u02<Q2\|{u_0}\|_{2}<\|{Q}\|_{2} lead to global in time solutions, where QQ is the ground state of the mass critical problem (ϵ=0\epsilon=0). We are interested by the threshold dynamic u02=Q2\|{u_0}\|_{2}=\|{Q}\|_{2} and in particular by the existence of finite time blow up minimal solutions. For ϵ=0\epsilon=0, such an object exists thanks to the explicit conformal symmetry, and is in fact unique from the seminal work [22]. For ϵ=1\epsilon=-1, simple variational arguments ensure that minimal mass data lead to global in time solutions. We investigate in this paper the case ϵ=1\epsilon=1, exhibiting a new class of minimal blow up solutions with blow up rates deeply affected by the double power nonlinearity. The analysis adapts the recent approach [31] for the construction of minimal blow up elements.

Cite this article

Stefan Le Coz, Yvan Martel, Pierre Raphaël, Minimal mass blow up solutions for a double power nonlinear Schrödinger equation. Rev. Mat. Iberoam. 32 (2016), no. 3, pp. 795–833

DOI 10.4171/RMI/899