This article deals with the following fractional Kirchhoff problem with critical exponent a + b ∫ R N ∣ ( − Δ ) s 2 u ∣ 2 d x ( − Δ ) s u = ( 1 + ε K ( x ) ) u 2 s ∗ − 1 , in R N , \left(a+b\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}| {\left(-\Delta )}^{\tfrac{s}{2}}u\hspace{-0.25em}{| }^{2}{\rm{d}}x\right){\left(-\Delta )}^{s}u=\left(1+\varepsilon K\left(x)){u}^{{2}_{s}^{\ast }-1},\hspace{1.0em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{N}, where a , b > 0 a,b\gt 0 are given constants, ε \varepsilon is a small parameter, 2 s ∗ = 2 N N − 2 s {2}_{s}^{\ast }=\frac{2N}{N-2s} with 0 < s < 1 0\lt s\lt 1 and N ≥ 4 s N\ge 4s . We first prove the nondegeneracy of positive solutions when ε = 0 \varepsilon =0 . In particular, we prove that uniqueness breaks down for dimensions N > 4 s N\gt 4s , i.e., we show that there exist two nondegenerate positive solutions which seem to be completely different from the result of the fractional Schrödinger equation or the low-dimensional fractional Kirchhoff equation. Using the finite-dimensional reduction method and perturbed arguments, we also obtain the existence of positive solutions to the singular perturbation problems for ε \varepsilon small.

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关于奇异摄动分数基尔霍夫方程：临界情况

本文处理以下具有临界指数的分数基尔霍夫问题 一种 + b ∫ R ñ ∣ ( - Δ ) s 2 你 ∣ 2 d X ( - Δ ) s 你 = ( 1 + ε ķ ( X ) ) 你 2 s * - 1 , 在 R ñ , \left(a+b\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}| {\left(-\Delta )}^{\tfrac{s}{2}} u\hspace{-0.25em}{| }^{2}{\rm{d}}x\right){\left(-\Delta )}^{s}u=\left(1+\varepsilon K\ left(x)){u}^{{2}_{s}^{\ast }-1},\hspace{1.0em}\hspace{0.1em}\text{in}\hspace{0.1em}\ hspace{0.33em}{{\mathbb{R}}}^{N}, 在哪里 一种 , b > 0 a,b\gt 0 给定常数， ε \伐普西隆 是一个小参数， 2 s * = 2 ñ ñ - 2 s {2}_{s}^{\ast }=\frac{2N}{N-2s} 和 0 < s < 1 0\lt s\lt 1 和 ñ ≥ 4 s N\ge 4s . 我们首先证明正解的非退化性，当 ε = 0 \伐普西隆 =0 . 特别是，我们证明了维度的唯一性分解 ñ > 4 s N\gt 4s ，即我们证明存在两个非退化正解，它们似乎与分数薛定谔方程或低维分数基尔霍夫方程的结果完全不同。使用有限维约简方法和扰动参数，我们还获得了奇异扰动问题的正解的存在性 ε \伐普西隆 小的。