Revista Matemática Complutense ( IF 0.8 ) Pub Date : 2021-02-23 , DOI: 10.1007/s13163-021-00388-w Maoding Zhen , Binlin Zhang
In this paper, we study normalized ground states for the following critical fractional NLS equation with prescribed mass:
$$\begin{aligned} {\left\{ \begin{array}{ll} (-\Delta )^{s}u=\lambda u +\mu |u|^{q-2}u+|u|^{2_{s}^{*}-2}u,&{}x\in \mathbb {R}^{N}, \\ \int _{\mathbb {R}^{N}}u^{2}dx=a^{2},\\ \end{array}\right. } \end{aligned}$$where \((-\Delta )^{s}\) is the fractional Laplacian, \(0<s<1\), \(N>2s\), \(2<q<2_{s}^{*}=2N/(N-2s)\) is a fractional critical Sobolev exponent, \(a>0\), \(\mu \in \mathbb {R}\). By using Jeanjean’s trick in Jeanjean (Nonlinear Anal 28:1633–1659, 1997), and the standard method which can be found in Brézis and Nirenberg (Commun Pure Appl Math 36:437–477, 1983) to overcome the lack of compactness, we first prove several existence and nonexistence results for a \(L^{2}\)-subcritical (or \(L^{2}\)-critical or \(L^{2}\)-supercritical) perturbation \(\mu |u|^{q-2}u\), then we give some results about the behavior of the ground state obtained above as \(\mu \rightarrow 0^{+}\). Our results extend and improve the existing ones in several directions.
中文翻译：
具有扰动的临界分数NLS方程的归一化基态
在本文中，我们研究具有规定质量的以下临界分数NLS方程的归一化基态：
$$ \ begin {aligned} {\ left \ {\ begin {array} {ll}（-\ Delta）^ {s} u = \ lambda u + \ mu | u | ^ {q-2} u + | u | ^ {2_ {s} ^ {*}-2} u，＆{} x \ in \ mathbb {R} ^ {N}，\\ \ int _ {\ mathbb {R} ^ {N}} u ^ { 2} dx = a ^ {2}，\\ \ end {array} \ right。} \ end {aligned} $$其中\（（-\ Delta）^ {s} \）是分数拉普拉斯算子，\（0 <s <1 \），\（N> 2s \），\（2 <q <2_ {s} ^^ ** } = 2N /（N-2s）\）是分数临界Sobolev指数\（a> 0 \），\（\ mu \ in \ mathbb {R} \）。通过使用Jeanjean中的Jeanjean技巧（非线性分析28：1633–1659，1997），以及在Brézis和Nirenberg中找到的标准方法（Commun Pure Appl Math 36：437–477，1983），可以克服紧凑性的不足，我们首先证明\（L ^ {2} \）-亚临界（或\（L ^ {2} \）-临界或\（L ^ {2} \）-超临界）摄动\（ \ mu | u | ^ {q-2} u \），然后给出一些关于上面获得的基态行为的结果，即\（\ mu \ rightarrow 0 ^ {+} \）。我们的结果在多个方向上扩展和改进了现有的结果。
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