The purpose of this paper is to study the nonlocal elliptic equation involving critical Hardy–Sobolev exponents as follows,$$\begin{aligned}(\mathrm{P}) {\left\{ \begin{array}{ll} (-\Delta )^s u -\mu \frac{u}{|x|^{2s}}= \lambda |u|^{q-2}u +\frac{|u|^{2_\alpha ^*-2}u}{|x|^\alpha } &{} \text {in} \ \Omega ,\\ u=0 &{} \text {in} \ \mathbb {R}^n\setminus \Omega , \end{array}\right. } \end{aligned}$$where \(\Omega \subset \mathbb {R}^N\) is a bounded domain with Lipschitz boundary, \(0<s<1\), \(\lambda >0\) is a parameter, \(0\le \mu <\mu _0,\) with \(\mu _0=4^s\frac{\Gamma ^2\left( \frac{N+2s}{4}\right) }{\Gamma ^2\left( \frac{N-2s}{4}\right) }\) being the sharp constant of the fractional Hardy–Sobolev in \({\mathbb R}^N,\)\(0< \alpha<2s<N\), \(1<q <2_s^*\) where \(2_s^* = \frac{2N}{N-2s}\) and \(2_\alpha ^* = \frac{2(N-\alpha )}{N-2s}\) are the fractional critical Sobolev and Hardy–Sobolev exponents respectively. The fractional Laplacian \((-\Delta )^s \) with \(s \in (0,1)\) is the non linear non local operator defined on smooth functions by:$$\begin{aligned} (-\Delta )^s u(x)=-\frac{1}{2} \int _{\mathbb {R}^N} \frac{u(x+y)+u(x-y)-2u(x)}{|y|^{N+2s}}\,\mathrm{d}y,\quad \text {for all } x \in \mathbb {R}^N. \end{aligned}$$We combine sub and super-solution method combine with min-max method in order to prove the existence and multiplicity of solutions to the problem \((\mathrm{P}).\)

中文翻译：

---

具有临界Sobolev–Hardy非线性的非局部问题的解的存在性和多重性

本文的目的是研究涉及临界Hardy-Sobolev指数的非局部椭圆方程，如下所示：$$ \ begin {aligned}（\ mathrm {P}）{\ left \ {\ begin {array} {ll}（- \ Delta）^ su-\ mu \ frac {u} {| x | ^ {2s}} = \ lambda | u | ^ {q-2} u + \ frac {| u | ^ {2_ \ alpha ^ *- 2} u} {| x | ^ \ alpha}＆{} \ text {in} \ \ Omega，\\ u = 0＆{} \ text {in} \ \ mathbb {R} ^ n \ setminus \ Omega， \ end {array} \ right。} \ end {aligned} $$其中\（\ Omega \ subset \ mathbb {R} ^ N \）是一个具有Lipschitz边界的有界域，\（0 <s <1 \），\（\ lambda> 0 \）是参数\（0 \ le \ mu <\ mu _0，\）与\（\ mu _0 = 4 ^ s \ frac {\ Gamma ^ 2 \ left（\ frac {N + 2s} {4} \ right ）} {\ Gamma ^ 2 \ left（\ frac {N-2s} {4} \ right）} \）作为分数次Hardy-的Sobolev在尖锐恒定\（{\ mathbb R} ^ N，\）\（0 <\阿尔法<2S <N \） ，\（1 <Q <2_S ^ * \） ，其中\（ 2_s ^ * = \ frac {2N} {N-2s} \）和\（2_ \ alpha ^ * = \ frac {2（N- \ alpha}} {N-2s} \）是分数临界Sobolev和Hardy – Sobolev指数。带\（s \ in（0,1）\）的分数拉普拉斯\\（（-\ Delta）^ s \）是在光滑函数上定义的非线性非局部算子，其计算方法如下：$$ \ begin {aligned}（-\ Delta）^ su（x）=-\ frac {1} {2} \ int _ {\ mathbb {R} ^ N} \ frac {u（x + y）+ u（xy）-2u（x）} { | y | ^ {N + 2s}} \，\ mathrm {d} y，\ quad \ text {对于所有} x \ in \ mathbb {R} ^ N。\ end {aligned} $$为了证明问题\（（\ mathrm {P}）。\）的解的存在性和多重性，我们将子解和超解法与min-max方法结合起来。