Global well-posedness and scattering for nonlinear Schr\"odinger equations with combined nonlinearities in the radial case

By Xing Cheng, Changxing Miao, Lifeng Zhao. 2014

67 citations recorded by the source corpus.

We consider the Cauchy problem for the nonlinear Schrodinger equation with combined nonlinearities, one of which is defocusing mass-critical and the other is focusing energy-critical or energy-subcritical. The threshold is given by means of variational argument. We establish the profile decomposition in $H^1(\Bbb R^d)$ and then utilize the concentration-compactness method to show the global wellposedness and scattering versus blowup in $H^1(\Bbb R^d)$ below the threshold for radial data when $d\leq4$.

Subfields: Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/b37ba7f29f166c0ac9f3c37aea33ae3316c560ff

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In Statistical and Nonlinear Physics

Citations in the corpus

Multiple normalized solutions for a Sobolev critical Schrödinger equation (2020, background): “s-Nirenberg [15], the existence of positive real solutions for equations of the type of (1.2) is addressed in a very general setting; to [1,2] which concerns the case where q>2 + 4/Nand µ>0; to [17,32] where the ﬁxed λ∈ Rproblem is analyzed for q= 2+4/Nand µ<0; to [26] where is studied the ”
Normalized solutions for a Schrödinger equation with critical growth in RN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{documen (2021, unknown): “In the purely L2-critical case, that is, q = 2 + 4 N , related problems were studied in [16,32].”
Orbital stability of ground states for a Sobolev critical Schrödinger equation (2020, background): “2) is addressed in a very general setting; to [1, 2] which concerns the case where q > 2 + 4/N and μ > 0; to [15, 29] where the fixed λ ∈ R problem is analyzed for q = 2+4/N and μ < 0; see also [26] and the reference therein.”
Normalized ground states for the NLS equation with combined nonlinearities: The Sobolev critical case (2019, unknown): “In this direction, we refer to [5,26], where the existence of positive decaying real solutions for elliptic equations in R is addressed in a very general setting; to [2,3], which concern the case when q > 2 + 4/N and μ > 0; to [25, 38], where the variational framework for the fixed λ problem is anal”
Normalized ground states for the NLS equation with combined nonlinearities (2018, unknown): “Global well-posedness, scattering, the occurrence of blow-up and more in general dynamical properties has been studied in [49] and many papers [2, 19, 22, 24, 29, 33, 36, 39, 40, 52] (see also the references therein).”