On the blow-up solutions for the nonlinear Schr\"{o}dinger equation with combined power-type nonlinearities

Abstract

This paper is devoted to the analysis of blow-up solutions for the nonlinear Schr\"{o}dinger equation with combined power-type nonlinearities \[ iu_{t}+\Delta u=\lambda_1|u|^{p_1}u+\lambda_2|u|^{p_2}u. \] 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. (Comm. Partial Differential Equations 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.