The Cauchy problem for the critical inhomogeneous nonlinear Schrödinger equation in Hˢ( Rⁿ)

In this paper, we study the Cauchy problem for the critical inhomogeneous nonlinear Schrödinger (INLS) equationiuₜ +Δ u = |x|⁻ᵇ f(u), \; u(0) = u₀ ∈ Hˢ ( Rⁿ ),where n≥3, 1< s<n/2, $ 0<b<2 $ and $ f(u) $ is a nonlinear function that behaves like λ |u|σ u with λ ∈ C and σ = 4-2b/n-2s. We establish the local well-posedness as well as the small data global well-posedness and scattering in Hˢ ( Rⁿ ) with 1<s<n/2 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.