New Blow-up Criteria And Stability Of Standing Waves For Nonhomogeneous Nonlinear Schr（？）dinger Equation

In this paper,we are conserned the existence of blow-up solutions and the orbital stability of standing waves to the following Schrodinger equation iut+Δu-V(x)u+|x|-b|u|2σ u=0,x∈RN,t≥0,where V(x)=∑j=1N aj2xj2,aj∈R.When a1=a2=…=aN=0,This paper establishs a sufficient condition for the existence of blow up solutions for nonhomogeneous nonlinear Schrodinger equation,under the condition of L2 supercritical,by constructing a class of invariant set and using the Gagliardo-Nirenberg type inequality and we prove that for any largeμ,there exists u0 ∈ H1 such that E(u0)=μ,and the solution u(t,x)with u0 as initial value blows up in finite time.This result improves the result of[15].When V(x)=∑j=1k aj2xj2(1≤k<),Nonhomogeneous nonlinear Schrodinger equation with a partial harmonic potential is considered.By using variational ar-guments,concentrated compactness lemma and the minimization sequence of the corresponding minimization problem,the existence and orbital stability of standing waves in L2-subcritical case are established.