Strong Instability Of Standing Waves For An INLS With Inverse-square Potential

The inhomogeneous nonlinear Schr(?)dinger equation with inverse-square potential has important significance in the study of quantum field equations,some black hole solutions of Einstein equations and the propagation of waves in nonlinear media and so on.In this paper,we discuss the strong instability of standing waves for an inhomogeneous Schr(?)dinger equation with inverse-square potential.By using Weinstein functional and the concentrationcompactness principle,the existence of standing waves is obtained.In the mass-critical case,applying the pseudoconformal transformation law,a blowup solution of equation with the positive energy is constructed by the ground state solution.And then,we get the standing waves is strongly unstable.In the supercritical case,by analyzing the structure of the equation,we construct the proper functionals and the corresponding cross-constrained variational problem.Then,the cross-invariant manifolds of the evolution flow are established,and hence,the sufficient condition for the existence of blowup solution is obtained.Thus,we have the strong instability of standing waves.