Arkiv för Matematik

Volume 61 (2023)

Number 2

Minimal-mass blow-up solutions for inhomogeneous nonlinear Schrödinger equations with growing potentials

Pages: 413 – 436



Naoki Matsui (Department of Mathematics, Tokyo University of Science, Tokyo, Japan)


In this paper, we consider the following equation:\[i \frac{\partial u}{\partial t} + \Delta u + g(x) {\lvert u \rvert}^{4/N} u - Wu = 0 \textrm{.}\]We construct a critical-mass solution that blows up at a finite time and describe the behaviour of the solution in the neighbourhood of the blow-up time. Banica–Carles–Duyckaerts (2011) have shown the existence of a critical-mass blow-up solution under the assumptions that $N \gt 2$, that $g$ and $W$ are sufficiently smooth and that each derivative of these is bounded. In this paper, we show the existence of a critical-mass blow-up solution under weaker assumptions regarding smoothness and boundedness of $g$ and $W$. In particular, it includes the cases where $W$ is unbounded at spatial infinity or not Lipschitz continuous.

Received 21 August 2022

Received revised 20 October 2022

Accepted 1 February 2023

Published 13 November 2023