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 ‐norm on 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 ‐norm on by using potential well theory, variational method, and some transformation techniques.