Gobal Well-posedness Of Several Classes Of Nonlinear Potential Schr(o|¨)dinger Equations

This paper discusses global well-posedness of several classes of nonlinear potentialSchro¨dinger equations.Firstly, we study a class of nonlinear Scho¨dinger equation with combined power-type non-linearities of same sign and specific harmonic potential. The characteristics is that this equationhas the combined power-type nonlinearities of same signa and harmonic potential. In additionthat one power of the nonlinear source term is critical and the other is supercritical, which givesus some difficult to deal with this complex potential well structure. In order to resolve thisproblem we construct a cross-constrained variational problem and so-called invariant manifoldsunder the flow of this problem and derive a sharp criterion for blow-up and global existence ofthe solutions.Secondly, we are interested in a class of nonlinear Schoo¨dinger equation with combinedpower-type nonlinearities of different sign and specific harmonic potential. The difficult of han-dling this equation is that the coefficient of the nonlinear source term is not fixed but variable.The way to overcome this issue is that by building up some spaces and functionals and con-structing a constrained variational problem and so-called invariant manifolds of the evolutionflow, we prove the existence of global solution and non-global solution. Furthermore by intro-ducing a family of potential wells and define some invariant sets under the flow, we derive asharp condition for blow-up and global existence of the solution.Finaly, we discuss a class of nonlinear Scho¨dinger equation with combined power-typenonlinearities of two different signs and generalized harmonic potential. The difference amongthis equation and the other same equations is that this considered equation processes the com-bined power nonlinear source terms, which makes the structure of this equation more generaland complex. Furthermore we generalize the harmonic potential to the general situation, thatmakes the discussion of solution more stylized. By introducing some spaces together with thepotential well, we derive a sharp criterion for blow-up and global existence of the solutions.Meanwhile we got the influence that the nonlinear source terms with two different signs haveon the soulutions of this problem and generalized this to those with combined different signs.