Abstract In this paper we investigate the following nonlocal problem with singular term and critical Hardy-Sobolev exponent (P)(−Δ)su=λuγ+|u|2α∗−2u|x|α in Ω,u>0 in Ω,u=0 in RN∖Ω, $$\begin{array}{} ({\rm P}) \left\{ \begin{array}{ll} (-\Delta)^s u = \displaystyle{\frac{\lambda}{u^\gamma}+\frac{|u|^{2_\alpha^*-2}u}{|x|^\alpha}} \ \ \text{ in } \ \ \Omega, \\ u >0 \ \ \text{ in } \ \ \Omega, \quad u = 0 \ \ \text{ in } \ \ \mathbb{R}^{N}\setminus \Omega, \end{array} \right. \end{array}$$ where Ω ⊂ ℝN is an open bounded domain with Lipschitz boundary, 0 < s < 1, λ > 0 is a parameter, 0 < α < 2s < N, 0 < γ < 1 < 2 < 2s∗ $\begin{array}{} \displaystyle 2_s^* \end{array}$, where 2s∗=2NN−2s and 2α∗=2(N−α)N−2s $\begin{array}{} \displaystyle 2_s^* = \frac{2N}{N-2s} ~\text{and}~ 2_\alpha^* = \frac{2(N-\alpha)}{N-2s} \end{array}$ are the fractional critical Sobolev and Hardy Sobolev exponents respectively. The fractional Laplacian (–Δ)s with s ∈ (0, 1) is the nonlinear nonlocal operator defined on smooth functions by (−Δ)su(x)=−12∫RNu(x+y)+u(x−y)−2u(x)|y|N+2sdy, for all x∈RN. $$\begin{array}{} \displaystyle (-\Delta)^s u(x)=-\frac{1}{2} \displaystyle\int_{\mathbb{R}^N} \frac{u(x+y)+u(x-y)-2u(x)}{|y|^{N+2s}}{\rm d }y, \;\; \text{ for all }\, x \in \mathbb{R}^N. \end{array}$$ By combining variational and approximation methods, we provide the existence of two positive solutions to the problem (P).

中文翻译：

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具有临界 Sobolev-Hardy 和奇异非线性的非局部偏微分方程的多重正解通过微扰法

摘要 在本文中，我们研究了以下具有奇异项和临界 Hardy-Sobolev 指数 (P)(−Δ)su=λuγ+|u|2α∗−2u|x|α in Ω,u>0 in Ω 的非局部问题， u=0 in RN∖Ω, $$\begin{array}{} ({\rm P}) \left\{ \begin{array}{ll} (-\Delta)^su = \displaystyle{\frac{ \lambda}{u^\gamma}+\frac{|u|^{2_\alpha^*-2}u}{|x|^\alpha}} \ \ \ text{ in } \ \ \Omega, \ \ u >0 \ \ \text{ in } \ \ \Omega, \quad u = 0 \ \ \text{ in } \ \ \ mathbb{R}^{N}\setminus \Omega, \end{array} \正确的。\end{array}$$ 其中 Ω ⊂ ℝN 是一个具有 Lipschitz 边界的开有界域，0 < s < 1，λ > 0 是一个参数，0 < α < 2s < N，0 < γ < 1 < 2 < 2s ∗ $\begin{array}{} \displaystyle 2_s^* \end{array}$, 其中 2s∗=2NN−2s 和 2α∗=2(N−α)N−2s $\begin{array}{} \displaystyle 2_s^* = \frac{2N}{N-2s} ~\text{and} ~ 2_\alpha^* = \frac{2(N-\alpha)}{N-2s} \end{array}$ 分别是分数临界 Sobolev 和 Hardy Sobolev 指数。具有 s ∈ (0, 1) 的分数拉普拉斯算子 (–Δ)s 是由 (−Δ)su(x)=−12∫RNu(x+y)+u(x−y) 在平滑函数上定义的非线性非局部算子)−2u(x)|y|N+2sdy，对于所有 x∈RN。$$\begin{array}{} \displaystyle (-\Delta)^su(x)=-\frac{1}{2} \displaystyle\int_{\mathbb{R}^N} \frac{u(x) +y)+u(xy)-2u(x)}{|y|^{N+2s}}{\rm d }y, \;\; \text{ 对所有 }\, x \in \mathbb{R}^N。\end{array}$$ 通过结合变分和近似方法，我们提供了问题 (P) 的两个正解的存在。1) 是由 (−Δ)su(x)=−12∫RNu(x+y)+u(x−y)−2u(x)|y|N+2sdy 在平滑函数上定义的非线性非局部算子，对于所有 x∈RN。$$\begin{array}{} \displaystyle (-\Delta)^su(x)=-\frac{1}{2} \displaystyle\int_{\mathbb{R}^N} \frac{u(x) +y)+u(xy)-2u(x)}{|y|^{N+2s}}{\rm d }y, \;\; \text{ 对所有 }\, x \in \mathbb{R}^N。\end{array}$$ 通过结合变分和近似方法，我们提供了问题 (P) 的两个正解的存在。1) 是由 (−Δ)su(x)=−12∫RNu(x+y)+u(x−y)−2u(x)|y|N+2sdy 在平滑函数上定义的非线性非局部算子，对于所有 x∈RN。$$\begin{array}{} \displaystyle (-\Delta)^su(x)=-\frac{1}{2} \displaystyle\int_{\mathbb{R}^N} \frac{u(x) +y)+u(xy)-2u(x)}{|y|^{N+2s}}{\rm d }y, \;\; \text{ 对所有 }\, x \in \mathbb{R}^N。\end{array}$$ 通过结合变分和近似方法，我们提供了问题 (P) 的两个正解的存在。