Remark on the Concentration Phenomenon for the Nonlinear Schrödinger Equations with a Repulsive Potential

Abstract

In this paper, we study the blow-up solutions for the \(L^2\)-supercritical nonlinear Schrödinger equation with a repulsive harmonic potential. In terms of the new sharp Gagliardo–Nirenberg inequality proposed by Weinstein (Commun PDE 11:545–565, 1986), we obtain the \({\dot{H}}^{s_c}\)-concentration phenomenon of blow-up solutions for this \(L^2\)-supercritical nonlinear Schrödinger equation in the space dimension \(N=2,3,4\).