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The existence and multiplicity of the normalized solutions for fractional Schrödinger equations involving Sobolev critical exponent in the L 2-subcritical and L 2-supercritical cases
Advances in Nonlinear Analysis ( IF 4.2 ) Pub Date : 2022-05-23 , DOI: 10.1515/anona-2022-0252
Quanqing Li 1 , Wenming Zou 2
Affiliation  

This paper is devoted to investigate the existence and multiplicity of the normalized solutions for the following fractional Schrödinger equation: (P) ( Δ ) s u + λ u = μ u p 2 u + u 2 s 2 u , x R N , u > 0 , R N u 2 d x = a 2 , \left\{\begin{array}{l}{\left(-\Delta )}^{s}u+\lambda u=\mu | u{| }^{p-2}u+| u{| }^{{2}_{s}^{\ast }-2}u,\hspace{1em}x\in {{\mathbb{R}}}^{N},\hspace{1.0em}\\ u\gt 0,\hspace{1em}\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}| u{| }^{2}{\rm{d}}x={a}^{2},\hspace{1.0em}\end{array}\right. where 0 < s < 1 0\lt s\lt 1 , a a , μ > 0 \mu \gt 0 , N 2 N\ge 2 , and 2 < p < 2 s 2\lt p\lt {2}_{s}^{\ast } . We consider the L 2 {L}^{2} -subcritical and L 2 {L}^{2} -supercritical cases. More precisely, in L 2 {L}^{2} -subcritical case, we obtain the multiplicity of the normalized solutions for problem ( P ) \left(P) by using the truncation technique, concentration-compactness principle, and genus theory. In L 2 {L}^{2} -supercritical case, we obtain a couple of normalized solution for ( P ) \left(P) by using a fiber map and concentration-compactness principle. To some extent, these results can be viewed as an extension of the existing results from Sobolev subcritical growth to Sobolev critical growth.

中文翻译:

在 L 2 亚临界和 L 2 超临界情况下涉及 Sobolev 临界指数的分数阶薛定谔方程的归一化解的存在性和多重性

本文致力于研究以下分数薛定谔方程的归一化解的存在性和多重性: (P) ( - Δ ) s + λ = μ p - 2 + 2 s * - 2 , X R ñ , > 0 , R ñ 2 d X = 一个 2 , \left\{\begin{array}{l}{\left(-\Delta )}^{s}u+\lambda u=\mu | 你{| }^{p-2}u+| 你{| }^{{2}_{s}^{\ast }-2}u,\hspace{1em}x\in {{\mathbb{R}}}^{N},\hspace{1.0em}\\ u\gt 0,\hspace{1em}\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}| 你{| }^{2}{\rm{d}}x={a}^{2},\hspace{1.0em}\end{array}\right. 在哪里 0 < s < 1 0\lt s\lt 1 , 一个 一个 , μ > 0 \亩\gt 0 , ñ 2 N\ge 2 , 和 2 < p < 2 s * 2\lt p\lt {2}_{s}^{\ast } . 我们认为 大号 2 {L}^{2} -亚临界和 大号 2 {L}^{2} - 超临界情况。更准确地说,在 大号 2 {L}^{2} -亚临界情况,我们获得问题的归一化解的多重性 ( ) \左(P) 通过使用截断技术、集中紧致原理和属理论。在 大号 2 {L}^{2} -超临界情况,我们获得了几个归一化的解决方案 ( ) \左(P) 通过使用纤维图和浓度紧凑性原理。在某种程度上,这些结果可以看作是从 Sobolev 亚临界增长到 Sobolev 临界增长的现有结果的延伸。
更新日期:2022-05-23
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