Local well‐posedness of a critical inhomogeneous Schrödinger equation

Abstract

In this note, one studies the inhomogeneous Schrödinger equation itrueu˙−false(−normalΔfalse)su=±false|xfalse|bfalse|ufalse|p−1u,0.30em0<s<1<p.$$ i\dot{u}-{\left(-\Delta \right)}&#x0005E;su&#x0003D;\pm {\left&#x0007C;x\right&#x0007C;}&#x0005E;b{\left&#x0007C;u\right&#x0007C;}&#x0005E;{p-1}u,\kern0.30em 0&lt;s&lt;1&lt;p. $$ Indeed, the local existence of solutions is established for a data u0∈Hμ$$ {u}_0\in {H}&#x0005E;{\mu } $$, where μ≥sc$$ \mu \ge {s}_c $$ and sc$$ {s}_c $$ is the Sobolev critical exponent given by the equality false‖λb+2sp−1u0false(λ·false)false‖trueH˙sc=false‖u0false‖trueH˙sc$$ {\left\Vert {\lambda}&#x0005E;{\frac{b&#x0002B;2s}{p-1}}{u}_0\left(\lambda \cdotp \right)\right\Vert}_{{\dot{H}}&#x0005E;{s_c}}&#x0003D;{\left\Vert {u}_0\right\Vert}_{{\dot{H}}&#x0005E;{s_c}} $$. In particular, one considers the mass‐critical regime: sc=0$$ {s}_c&#x0003D;0 $$ and the energy critical regime: s=sc$$ s&#x0003D;{s}_c $$. In order to use Strichartz estimates without loss of regularity, one considers spherically symmetric data. To the authors' knowledge, the local well‐posedness of the inhomogeneous fractional Schrödinger equation (FINLS) in the critical Sobolev spaces remains open. In fact, the method used in proving the existence of solutions in the subcritical regime is no more applicable in the critical one. For more efficiency to handle the spatially decaying factor false|xfalse|b$$ {\left&#x0007C;x\right&#x0007C;}&#x0005E;b $$ in the source term, we approach to the matter in a weighted Lebesgue space which seems to be more suitable to perform a finer analysis for this problem. The novelty here is to consider the critical regime. This works follows some ideas which treat the Laplacian case. The non‐local fractional Laplacian gives some serious complications and make the problem more difficult. The present study is a natural extension of the existing literature about the well‐posedness of the FINLS in Sobolev spaces.

Local well-posedness of the fractional inhomogeneous NLS in Sobolev spaces

“…• The method in this paper which consists on using the integrity of |x| −τ in L r (A), where A ∈ {B, B c } fails in the H λ critical regime. • In the energy-critical regime, a local theory was developed by the third author [25].…”

“…The scattering of the inter-critical radial solutions was established by the third author [18] by use of Dodson-Murphy method [7]. In the energy-critical regime, the last author developed a local theory in Sobolev spaces [25] and obtained a decay result in the mass-sub-critical regime [24]. Using Lorentz spaces, the local well-posedness of the inhomogeneous Schrödinger equation with a mixed fractional Laplacian, was established in [3].…”