We studied the Cauchy problem for the inhomogeneous nonlinear Schrödinger-type equations:
where k=1,2, b>0, σ>0.
The inhomogeneous nonlinear Schrödinger-type equations appear in diverse branches of physics such as in nonlinear optics.
The case b=0 corresponds to the classical nonlinear Schrödinger-type equations which has been widely studied over the last three decades.
In the study of the classical nonlinear Schrödinger-type equations, the theory of Besov or Soblev spaces was mainly applied.
However, the theory of Besov or Sobolev spaces could be applied to the study of the inhomogeneous nonlinear Schrödinger-type equations only for the subcritical case.
Recently, Aloui-Tayachi [Discrete Contin. Dyn. Syst. 41(11) (2021) 5409–5437] gave a new strategy in order to establish the local well-posedness in Hs for both of critical and subcritical case.
More precisely, they used the theory of Sobolev-Lorentz spaces. However, they only considered the case s≤1. The restriction s≤1 comes from the fractional chain rule in Lorentz spaces obtained by them.
Motivated by this paper, we studied some properties such as various embeddings, interpolation inequality in Sobolev-Lorentz spaces. In particular, we obtained the generalized fractional chain rule in Lorentz spaces which holds for s≥0. Based on these results, we gave the unified approach to establish local and global well-posedness for the inhomogeneous nonlinear Schrödinger-type equations in the fractional Sobolev spaces Hs with s≥0.
The detailed results for the second-order case were published in "Zeitschrift für Analysis und ihre Anwendungen 42 (2023), 403–433" under the title of "A note on the Hs-critical inhomogeneous nonlinear Schrödinger equation"(https://doi.org/10.4171/ZAA/1745).
And the results for the fourth-order case were published in "Discrete and Continuous Dynamical Systems-Series B" under the title of "Sobolev-Lorentz spaces with an application to the inhomogeneous biharmonic NLS equation"(https://doi.org/10.3934/dcdsb.2024006).
