We consider the nonlocal Hénon equation

$$\begin{aligned} (-\Delta )^s u= |x|^{\alpha } u^{p},\quad {\mathbb {R}}^{N}, \end{aligned}$$

where \((-\Delta )^s\) is the fractional Laplacian operator with \(0<s<1\), \(-2s<\alpha \), \(p>1\) and \(N>2s\). We prove a nonexistence result for positive solutions in the optimal range of the nonlinearity, that is, when

$$\begin{aligned} 1<p<p^*_{\alpha , s}:=\frac{N+2\alpha +2s}{N-2s}. \end{aligned}$$

Moreover, we prove that a bubble solution, that is a fast decay positive radially symmetric solution, exists when \(p=p_{\alpha , s}^{*}\).

中文翻译：

---

$$ {\ mathbb {R}} ^ {N} $$ RN中非局部Hénon方程的研究中的尖锐指数：一个Liouville定理和一个存在结果

我们考虑非局部Hénon方程

$$ \ begin {aligned}（-\ Delta）^ su = | x | ^ {\\ alpha} u ^ {p}，\ quad {\ mathbb {R}} ^ {N}，\ end {aligned} $$

其中\（（-\ Delta）^ s \）是分数\\（0 <s <1 \），\（-2s <\ alpha \），\（p> 1 \）和\（N> 2s \）。我们证明了非线性最优范围内正解的不存在结果，即

$$ \ begin {aligned} 1 <p <p ^ * _ {\ alpha，s}：= \ frac {N + 2 \ alpha + 2s} {N-2s}。\ end {aligned} $$

此外，我们证明当\（p = p _ {\ alpha，s} ^ {*} \）时，存在气泡解，即快速衰减正径向对称解。