THRESHOLD SOLUTIONS FOR THE INTERCRITICAL INHOMOGENEOUS NLS
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.
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
Mathematics and Statistics
Keywords and Phrases
asymptotic behavior; blowup; nonlinear Schrödinger equation; scattering; threshold solutions
International Standard Serial Number (ISSN)
Article - Journal
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01 Jan 2023