THRESHOLD SOLUTIONS FOR THE INTERCRITICAL INHOMOGENEOUS NLS
Abstract
We consider the focusing inhomogeneous nonlinear Schrödinger equation in H1(R3), i∂ tu + Δ u + | x| b| u| 2u = 0, where 0 < b < 1 2 . Previous works (see, e.g., [L. Campos, Nonlinear Anal., 202 (2021), 112118; L. G. Farah and C. M. Guzmán, J. Differential Equations, 262 (2017), pp. 4175-4231; J. Murphy, Proc. Amer. Math. Soc., 150 (2022), pp. 1177-1186]) have established a blowup/scattering dichotomy below a mass-energy threshold determined by the ground state solution Q. In this work, we study solutions exactly at this mass-energy threshold. In addition to the ground state solution, we prove the existence of solutions Qpm , which approach the standing wave in the positive time direction but either blow up or scatter in the negative time direction. Using these particular solutions, we classify all possible behaviors for threshold solutions. In particular, the solution either behaves as in the subthreshold case, or it agrees with eitQ, Q+, or Q up to the symmetries of the equation.
Recommended Citation
L. Campos and J. Murphy, "THRESHOLD SOLUTIONS FOR THE INTERCRITICAL INHOMOGENEOUS NLS," SIAM Journal on Mathematical Analysis, vol. 55, no. 4, pp. 3807 - 3843, Society for Industrial and Applied Mathematics, Jan 2023.
The definitive version is available at https://doi.org/10.1137/22M1497663
Department(s)
Mathematics and Statistics
Keywords and Phrases
asymptotic behavior; blowup; nonlinear Schrödinger equation; scattering; threshold solutions
International Standard Serial Number (ISSN)
1095-7154; 0036-1410
Document Type
Article - Journal
Document Version
Final Version
File Type
text
Language(s)
English
Rights
© 2023 Society for Industrial and Applied Mathematics, All rights reserved.
Publication Date
01 Jan 2023
Comments
Fundação de Amparo à Pesquisa do Estado de São Paulo, Grant None