Threshold solutions for the intercritical inhomogeneous NLS

Luccas Campos; Jason Murphy

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arXiv · 2205.09714

Threshold solutions for the intercritical inhomogeneous NLS

Luccas Campos·Jason Murphy

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Abstract

We consider the focusing inhomogeneous nonlinear Schr\"odinger equation in $H^1(\mathbb{R}^3)$, \begin{equation} i\partial_t u + \Delta u + |x|^{-b}|u|^{2}u=0,{equation} where $0 < b <\tfrac{1}{2}$. Previous works 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 $Q^\pm$, 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 sub-threshold case, or it agrees with $e^{it}Q$, $Q^+$, or $Q^-$ up to the symmetries of the equation.

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Luccas Campos, Jason Murphy. 2022-05-19. Threshold solutions for the intercritical inhomogeneous NLS. https://arxiv.org/abs/2205.09714

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