摘要： In this article, we consider the following coupled fractional nonlinear Schr?dinger system in $\mathbb{R}^{N}$

$$\left\{ \begin{array}{l} {\left( { - \Delta } \right)^s}u + P\left( x \right)u = {\mu _1}{\left| u \right|^{2p - 2}}u + \beta {\left| u \right|^p}{\left| u \right|^{p - 2}}u,\;\;\;x \in {{\mathbb{R}}^N},\\{\left( { - \Delta } \right)^s}v + Q\left( x \right)v = {\mu _2}{\left| v \right|^{2p - 2}}v + \beta {\left| v \right|^p}{\left| v \right|^{p - 2}}v,\;\;\;\;\;x \in {{\mathbb{R}}^N},\\u,\;\;v \in {H^s}\left( {{{\mathbb{R}}^N}} \right),\end{array} \right.$$ where $N≥2, 0 < s < 1, 1 < p < \frac{N}{N-2s},\mu_1>0, \mu_2>0$ and $\beta \in \mathbb{R}$ is a coupling constant. We prove that it has infinitely many non-radial positive solutions under some additional conditions on $P(x), Q(x), p$ and $\beta$. More precisely, we will show that for the attractive case, it has infinitely many non-radial positive synchronized vector solutions, and for the repulsive case, infinitely many non-radial positive segregated vector solutions can be found, where we assume that $P(x)$ and $Q(x)$ satisfy some algebraic decay at infinity.

关键词: Hilbert problem 15, nonstandard analysis, enumeration geometry

Abstract: Hilbert problem 15 required understanding Schubert's book. In this book, reducing to degenerate cases was one of the main methods for enumeration. We found that nonstandard analysis is a suitable tool for making rigorous of Schubert's proofs of some results, which used degeneration method, but are obviously not rigorous. In this paper, we give a rigorous proof for Example 4 in Schubert's book, Chapter 1. §4 according to his idea. This shows that Schubert's intuitive idea is correct, but to make it rigorous a lot of work should be done.

Key words: Hilbert problem 15, nonstandard analysis, enumeration geometry

中图分类号: 

• 14N15