Blow-up solutions of the intercritical inhomogeneous NLS equation: the non-radial case

Abstract

In this paper we consider the inhomogeneous nonlinear Schrödinger (INLS) equation $$\begin{aligned} i \partial _t u +\Delta u +|x|^{-b} |u|^{2\sigma }u = 0, \,\,\, x \in {\mathbb {R}}^N \end{aligned}$$with \(N\ge 3\). We focus on the intercritical case, where the scaling invariant Sobolev index \(s_c=\frac{N}{2}-\frac{2-b}{2\sigma }\) satisfies \(0<s_c<1\). In a previous work, for radial initial data in \(\dot{H}^{s_c}\cap \dot{H}^1\), we prove the existence of blow-up solutions and also a lower bound for the blow-up rate. Here we extend these results to the non-radial case. We also prove an upper bound for the blow-up rate and a concentration result for general finite time blow-up solutions in \(H^1\).