Abstract
In this note, one studies the inhomogeneous Schrödinger equation
iu˙−(−Δ)su=±|x|b|u|p−1u,0<s<1<p.$$ i\dot{u}-{\left(-\Delta \right)}^su=\pm {\left|x\right|}^b{\left|u\right|}^{p-1}u,\kern0.30em 0<s<1<p. $$
Indeed, the local existence of solutions is established for a data
u0∈Hμ$$ {u}_0\in {H}^{\mu } $$, where
μ≥sc$$ \mu \ge {s}_c $$ and
sc$$ {s}_c $$ is the Sobolev critical exponent given by the equality
‖λb+2sp−1u0(λ·)‖H˙sc=‖u0‖H˙sc$$ {\left\Vert {\lambda}^{\frac{b+2s}{p-1}}{u}_0\left(\lambda \cdotp \right)\right\Vert}_{{\dot{H}}^{s_c}}={\left\Vert {u}_0\right\Vert}_{{\dot{H}}^{s_c}} $$. In particular, one considers the mass‐critical regime:
sc=0$$ {s}_c=0 $$ and the energy critical regime:
s=sc$$ s={s}_c $$. 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
|x|b$$ {\left|x\right|}^b $$ 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.