Spiked solutions for fractional Schrödinger systems with Sobolev critical exponent,Analysis and Mathematical Physics

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Spiked solutions for fractional Schrödinger systems with Sobolev critical exponent
Analysis and Mathematical Physics ( IF 1.7 ) Pub Date : 2024-02-26 , DOI: 10.1007/s13324-024-00878-2
Wenjing Chen , Xiaomeng Huang

In this article, we study the following fractional critical Schrödinger system

$$\begin{aligned} {\left\{ \begin{array}{ll} (-\Delta )^s u_i=\mu _iu_i^3+\beta u_i\sum _{j\ne i}u_j^{2}+\lambda _iu_i &{}\text { in } \ \Omega ,\\ u_i=0 &{}\text { on } \ {\mathbb {R}}^N\setminus \Omega , \end{array}\right. } \quad i=1,2,\ldots ,m, \end{aligned}$$

where $0<s<1$, $\mu _i>0$, coupling constant $\beta $ satisfies either $-\infty <\beta \le {\bar{\beta }}$ (${\bar{\beta }}>0$ small) or $\beta \rightarrow -\infty $, $0<\lambda _i<\lambda _1^s(\Omega )$, where $\lambda _1^s(\Omega )$ is the first eigenvalue of $(-\Delta )^s$ on $\Omega $, with $\Omega $ is a smooth bounded domain in ${\mathbb {R}}^N$ with $N=4s$. Under some geometric assumptions on $\Omega $, we construct solutions which concentrate and blow up at different points as $\lambda _1,\ldots ,\lambda _m\rightarrow 0$.

更新日期：2024-02-27

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