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A sharp Gagliardo-Nirenberg inequality and its application to fractional problems with inhomogeneous nonlinearity

The purpose of this paper is threefold. Firstly, we establish a Gagliardo-Nirenberg inequality with optimal constant, which involves a fractional norm and an inhomogeneous nonlinearity. Secondly, as an application of this inequality, we study ground state standing waves to a nonlinear Schrödinger equation (NLS) with a mixed fractional Laplacians and a inhomogeneous nonlinearity, and consider a minimization problem which gives the existence of ground state solutions with prescribed mass. In particular, by making use of this Gagliardo-Nirenberg inequality and its optimal constant, we give a sufficient and necessary condition for the existence results. Finally, we develop local wellposedness theory for NLS with a mixed fractional Laplacians and a inhomogeneous nonlinearity. In the process, we prove Strichartz estimates in Lorentz spaces which may be of independent interest.