On well-posedness for the inhomogeneous nonlinear Schrödinger equation in the critical case

Local well-posedness for the inhomogeneous nonlinear Schrödinger equation

“…5) The local well-posedness result in the critical case α = (4 − 2b)/(N − 2s) is completely new for N = 1, N = 2 and N ≥ 3 with 1/3 < s < 1. For the other cases, the solution constructed here is more regular than the one given in [29,30]. This follows by using the Hölder inequality in the Lorentz spaces.…”

“…252] and [14, line 38, p. 171], that the local well-posedness for the critical case α = (4 − 2b)/(N − 2s) is an open problem. Recently, we have learned in [29,30] that the local existence for this case is established using Strichartz estimates in some weighted Lebesgue spaces. Their results are given for N ≥ 3, 0 ≤ s < 1/3 or s = 1 and under some restrictive hypotheses on α and b.…”

“…See also [9] for radially symmetric initial data inḢ 1 (R 3 ). It seems that the spaces used in [20,13,29,30] are not the naturel one to give the complete local well-posedness theory. In fact, no blowup criterium is stated for the critical case.…”

“…Also, neither continuous dependence in H s in the standard sense (see [6] for b = 0) nor unconditional uniqueness are given even for the subcritical case. See [29,Theorem 1.1] and [20,Theorem 1.4] for continuous dependence results in weaker sense.…”

“…Our approach is to consider Strichartz estimates in Lorentz spaces. The solutions that we construct belong to the spaces considered in [20,13,29,30]. Moreover, we aim to build a local theory as the one developed for the case b = 0.…”