THE CAUCHY PROBLEM FOR THE CRITICAL INHOMOGENEOUS NONLINEAR SCHRO spacing diaeresis DINGER EQUATION IN Hs(Rn)

In this paper, we study the Cauchy problem for the critical inhomogeneous nonlinear Schro spacing diaeresis dinger (INLS) equationiut + Delta u = |x|-bf(u), u(0) = u0 is an element of Hs(Rn) where n >= 3, 1 < s < n2 , 0 < b < 2 and f(u) is a nonlinear function that behaves like lambda|u|Sigma u with lambda is an element of C and Sigma = 4-2b n-2s . We establish the local wellposedness as well as the small data global well-posedness and scattering in Hs(Rn) with 1 < s < n2 for the critical INLS equation under some assumption on b. To this end, we first establish various nonlinear estimates by using fractional Hardy inequality and then use the contraction mapping principle based on Strichartz estimates.