This paper deals with the following non-linear equation with a fractional Laplacian operator and almost critical exponents: \[ (-\Delta)^{s} u=K(|y'|,y'')u^{({N+2s})/(N-2s)\pm\epsilon},\quad u > 0,\quad u\in D^{1,s}(\mathbb{R}^{N}), \] where N ⩾ 4, 0 < s < 1, (y′, y″) ∈ ℝ2 × ℝN−2, ε > 0 is a small parameter and K(y) is non-negative and bounded. Under some suitable assumptions of the potential function K(r, y″), we will use the finite-dimensional reduction method and some local Pohozaev identities to prove that the above problem has a large number of bubble solutions. The concentration points of the bubble solutions include a saddle point of K(y). Moreover, the functional energies of these solutions are in the order $\epsilon ^{-(({N-2s-2})/({(N-2s)^2})}$.

中文翻译：

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具有几乎临界指数的分数阶椭圆方程的大量气泡解

本文处理以下具有分数拉普拉斯算子和几乎临界指数的非线性方程：\[ (-\Delta)^{s} u=K(|y'|,y'')u^{({N+2s})/(N-2s)\pm\epsilon},\quad u > 0,\quad u\in D^{1,s}(\mathbb{R}^{N}), \]在哪里ñ⩾ 4, 0 <s< 1, (是的',是的″) ∈ ℝ2× ℝñ-2, ε > 0 是一个小参数并且ķ(是的) 是非负的且有界的。在势函数的一些适当假设下ķ(r,是的”)，我们将使用有限维约简方法和一些局部 Pohozaev 恒等式来证明上述问题有大量的气泡解。气泡溶液的浓度点包括一个鞍点ķ(是的）。此外，这些溶液的功能能是有序的$\epsilon ^{-(({N-2s-2})/({(N-2s)^2})}$.