Abstract
In this paper, we study the critical norm conjecture for the inter-critical nonlinear Schrödinger equation with critical index sc satisfying 12<sc<1 when d≥5. Under the assumption of uniform boundedness of the critical norm, we prove the global well-posedness and scattering for the Cauchy problem. We follow the standard ‘Concentration compactness/Rigidity method’ established in [15,16], and treat three scenarios for the critical element respectively. Moreover, double Duhamel method and interaction Morawetz estimate are applied to exclude the critical element.
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