On the blow-up solutions for the nonlinear Schrödinger equation with combined power-type nonlinearities

Abstract

This paper is devoted to the analysis of blow-up solutions for the nonlinear Schrodinger equation with combined power-type nonlinearities $$\begin{aligned} iu_{t}+\Delta u=\lambda _1|u|^{p_1}u+\lambda _2|u|^{p_2}u. \end{aligned}$$ When $$p_1=\frac{4}{N}$$ and $$0<p_2<\frac{4}{N}$$ , we prove the existence of blow-up solutions and find the sharp threshold mass of blow-up and global existence for this equation. This is a complement to the result of Tao et al. (Commun Partial Differ Equ 32:1281–1343, 2007). Moreover, we investigate the dynamical properties of blow-up solutions, including $$L^2$$ -concentration, blow-up rates and limiting profile. When $$\frac{4}{N}<p_1<\frac{4}{N-2}$$ ( $$4<p_1<\infty $$ if $$N=1$$ , $$2<p_1<\infty $$ if $$N=2$$ ), we prove that the blow-up solution with bounded $$\dot{H}^{s_c}$$ -norm must concentrate at least a fixed amount of the $$\dot{H}^{s_c}$$ -norm and, also, its $$L^{p_c}$$ -norm must concentrate at least a fixed $$L^{p_c}$$ -norm.