Existence Of Blow-Up Solutions And Orbital Stability Of Standing Wave Solutions For Schr(?)dinger Equation Of Dipole Quantum Gas

In this paper,we study the existence of blow-up solutions and the orbital sta-bility of standing wave solutions for the following dipolar quantum gases Schrodinger equation where W(x)=a2(x12+x22+x32).When ?3=0,by constructing an invariant set,we establish a new blow-up criterion which implies the existence of blow-up solutions with arbitrarily large initial energy.This result gives a positive answer to the problem left by Carles,Markowich,and Sparber in[12].When ?3<0 and 0<p<4,we can prove the existence of stable standing waves for this equation by studying a local minimization problem.Our results generalize and improve the results in[27].