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. This paper contains two parts. In the first part, we derive a variant of GagliardoNirenberg 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 Schrödinger equation with harmonic potential and nonlocal nonlinearity

Sharp contrast in nonlocal inequality and its applications to nonlocal Schrödinger equation with harmonic potential