Sharp thresholds of blowup and uniform bound for a Schrödinger system with second-order derivative-type and combined power-type nonlinearities

Abstract

Considered herein is a Cauchy problem for a system of Schrödinger equations with second-order derivative-type and combined power-type nonlinearities. Through the effective combination of potential well theory, conservation laws, and vector-valued Gargliardo–Nirenberg inequality, we establish the uniform boundedness in $H$-norm on $[0,T)$ and corresponding decay rate estimate. Moreover, we also prove the existence of corresponding ground-state solutions for this problem. Finally, we mainly investigate three different sharp thresholds for blowup and uniform bound of solutions in $H$-norm on $[0,T)$ by using potential well theory, variational method, and some transformation techniques.