SHARP CONTRAST IN NONLOCAL INEQUALITY AND ITS APPLICATIONS TO NONLOCAL SCHRODINGER EQUATION WITH HARMONIC POTENTIAL

摘要

This paper contains two parts. In the first part, we derive a variant of Gagliardo-Nirenberg interpolation inequality involving nonlocal nonlinearity and determine its best (smallest) constant. In the second part, we study two applications of this inequality and its best constant. In the first application, we use this best constant to establish a sharp criterion for the global existence and blow-up of solutions of the inhomogeneous Schrodinger equation with harmonic potential and nonlocal nonlinearity i phi t = -Delta phi+vertical bar x vertical bar(2)phi-phi vertical bar phi vertical bar(p-2)integral vertical bar phi(y)vertical bar(p)/vertical bar x-y vertical bar(alpha)dy in the critical case p= 2+(2-alpha)/N. The result indicates that the existence of blow-up solutions not only depends on the mass of the initial data but also on the profile of the initial data. In the second application, we use this best constant to prove that when 2+(2-alpha)/N < p <(2N-alpha)/(N-2), the solutions exist globally in time for one class of initial data whose norm can be as large as one wants.