In this paper, we first establish Liouville type theorems for stable solutions in the whole space ${\mathbb{R}}^N$ to the fractional elliptic equation ${(-\Delta )}^{s}u=f\left(u\right),$where the nonlinearity is nondecreasing and convex. This extends some result in Dupaigne and Farina (2010) to the fractional setting. Our next purpose is to prove the nonexistence of stable positive solutions to the fractional Lane–Emden system $\left\{\begin{array}{c}{(-\Delta )}^{s}u=v^p\text{ in }{\mathbb{R}}^N\hfill \\ {(-\Delta )}^{s}v=u^q\text{ in }{\mathbb{R}}^N,\hfill \end{array}\right.$in the subcritical cases. This is, in particular, the first nonexistence result of stable positive solutions for the fractional Lane–Emden system in literature which extends the result in Cowan (2013) from the local case to the nonlocal one.

在本文中，我们首先建立了Liouville型定理，以求在整个空间中具有稳定的解 ${\mathbb{[R}}^{ñ}$ 到分数椭圆方程 ${（-\Delta ）}^sü=F（ü），$非线性是非递减的和凸的。这将Dupaigne和Farina（2010）的一些结果扩展到分数设置。我们的下一个目的是证明分数Lane-Emden系统不存在稳定的正解$\left\{\begin{array}{c}{（-\Delta ）}^sü=v^p\text{ 在 }{\mathbb{[R}}^{ñ}\hfill \\ {（-\Delta ）}^{s}v={ü}^q\text{ 在 }{\mathbb{[R}}^{ñ}，\hfill \end{array}\right.$在亚临界情况下。特别是，这是文献中关于分数Lane-Emden系统的稳定正解的第一个不存在结果，该结果将Cowan（2013）中的结果从本地案例扩展到了非本地案例。