On sharp scattering threshold for the mass-energy double critical NLS via double track profile decomposition

2021/08/02 by Yongming Luo, Luo, Yongming
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Black Holes and Theoretical Physics #FOS: Mathematics #Nonlinear Waves and Solitons

paper · pdf · doi:10.48550/arxiv.2108.00915

openalex publication_date 2021/08/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The present paper is concerned with the large data scattering problem for the mass-energy double critical NLS i∂_t u+Δu± |u|^(4)/(d)u± |u|^(4)/(d-2)u=0 in H¹(ℝ^d) with d≥ 3. In the defocusing-defocusing regime, Tao, Visan and Zhang show that the unique solution of DCNLS is global and scattering in time for arbitrary initial data in H¹(ℝ^d). This does not hold when at least one of the nonlinearities is focusing, due to the possible formation of blow-up and soliton solutions. However, precise thresholds for a solution of DCNLS being scattering were open in all the remaining regimes. Following the classical concentration compactness principle, we impose sharp scattering thresholds in terms of ground states for DCNLS in all the remaining regimes. The new challenge arises from the fact that the remainders of the standard L²- or H¹-profile decomposition fail to have asymptotically vanishing diagonal L²- and H¹-Strichartz norms simultaneously. To overcome this difficulty, we construct a double track profile decomposition which is capable to capture the low, medium and high frequency bubbles within a single profile decomposition and possesses remainders that are asymptotically small in both of the diagonal L²- and H¹-Strichartz spaces.

Citations

• Solitons and Scattering for the Cubic–Quintic Nonlinear Schrödinger Equation on ℝ³ R 3
• Uniqueness of positive solutions of Δu−u+up=0 in Rn
• Nonlinear scalar field equations, II existence of infinitely many solutions
• Nonlinear Schr�dinger equations and sharp interpolation estimates
• Existence and instability of standing waves with prescribed norm for a class of Schrödinger-Poisson equations
• Nonlinear scalar field equations, I existence of a ground state
• Normalized ground states for the NLS equation with combined nonlinearities: The Sobolev critical case

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