In this paper, we study the existence and asymptotic properties of solutions to the following fractional Schrödinger equation:

$${(-\mathrm{\Delta })}^{\sigma }u=\lambda u+|u{|}^{q-2}u+\mu \left(I_{\alpha }\ast |u{|}^p\right)|u{|}^{p-2}u\text{in}{ℝ}^N$$

under the normalized constraint

$${\int }_{{ℝ}^N}{u}^2=a^2,$$

where N ≥ 2, σ ∈ (0, 1), α ∈ (0, N), q ∈ (2 + 4σ/N, 2N/N − 2σ], p ∈ [2, 1 + 2σ + α/N), a > 0, μ > 0, $I_{\alpha }(x)=|x{|}^{\alpha -N}$ and $\lambda \in ℝ$ appears as a Lagrange multiplier. In the Sobolev subcritical case q ∈ (2 + 4σ/N, 2N/N − 2σ), we show that the problem admits at least two solutions under suitable assumptions on α, a, and μ. In the Sobolev critical case $q=2N/N-2\sigma$, we prove that the problem possesses at least one ground state solution. Furthermore, we give some stability and asymptotic properties of the solutions. We mainly extend the results of S. Bhattarai published in 2017 on J. Differ. Equ. and B. H. Feng et al published in 2019 on J. Math. Phys. concerning the above problem from L²-subcritical and L²-critical setting to L²-supercritical setting with respect to q, involving Sobolev critical case especially.

中文翻译：

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具有聚焦非局部扰动的分数阶薛定谔方程的归一化解

在本文中，我们研究了以下分数阶薛定谔方程解的存在性和渐近性质：

$${(-\mathrm{\Delta })}^{\sigma }你=\lambda 你+|你{|}^{q-2}你+\mu \left({\mathrm{一世}}_{\alpha }＊|你{|}^{磷}\right)|你{|}^{磷-2}你\text{在}{ℝ}^N$$

在规范化约束下

$${\int }_{{ℝ}^N}{你}^2={\mathrm{一种}}^2,$$

其中N ≥ 2, σ ∈ (0, 1), α ∈ (0,  N ), q ∈ (2 + 4 σ / N , 2 N / N  − 2 σ ], p ∈ [2, 1 + 2 σ  +  α / N ), a > 0, μ > 0,${\mathrm{一世}}_{\alpha }(X)=|X{|}^{\alpha -N}$ 和 $\lambda \in ℝ$表现为拉格朗日乘数。在 Sobolev 次临界情况q ∈ (2 + 4 σ / N , 2 N / N  − 2 σ ) 中，我们表明该问题在α、a和μ 的适当假设下至少有两个解。在 Sobolev 临界情况下$q=2N/N-2\sigma$，我们证明该问题至少有一个基态解。此外，我们给出了解的一些稳定性和渐近性质。我们主要扩展了 S. Bhattarai 于 2017 年在 J.Differ 上发表的结果。设备 和 BH Feng 等人于 2019 年发表在 J. Math 上。物理。关于上述从L ^{2 -}亚临界和L ^{2 -}临界设定到L ^{2 -}超临界设定关于q的问题，特别涉及Sobolev临界情况。