L 2 Concentration of Blow-Up Solutions for the Nonlinear Schrödinger Equation with an Inhomogeneous Combined Non-Linearity

Baoli Xie (), Congming Peng and Caochuan Ma
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Baoli Xie: School of Mathematics and Statistics, Tianshui Normal University, Tianshui 741000, China
Congming Peng: School of Mathematics and Statistics, Tianshui Normal University, Tianshui 741000, China
Caochuan Ma: School of Mathematics and Statistics, Tianshui Normal University, Tianshui 741000, China

Mathematics, 2024, vol. 12, issue 7, 1-16

Abstract: This article studies the Schrödinger equation with an inhomogeneous combined term i ∂ t u − ( − Δ ) s u + λ 1 | x | − b | u | p u + λ 2 | u | q u = 0 , where s ∈ ( 1 2 , 1 ) , λ 1 , λ 2 = ± 1 , 0 < b < { 2 s , N } and p , q > 0 . We study the limit behaviour of the infinite blow-up solution at the blow-up time. When the parameters p , q , λ 1 and λ 2 have different values, we obtain the nonexistence of a strong limit for the non-radial solution and the L 2 concentration for the radial solution. Interestingly, we find that the mass of the finite time blow-up solutions are concentrated in different ways for different parameters.

Keywords: inhomogeneous Schrödinger equation; L2 concentration; limit behaviour (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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