Using the Mountain-Pass Theorem of Ambrosetti and Rabinowitz we prove that the following fractional $p$-Laplacian equation with double critical nonlinearities ${(-{\Delta }_p)}^{s}u={\left|u\right|}^{p_s^{\ast }-2}u+\frac{{\left|u\right|}^{p_s^{\ast }\left(\alpha \right)-2}u}{{\left|x\right|}^{\alpha }}in{\mathbb{R}}^N,$admits a positive solution in the class ${\dot{W}}^{s,p}\left({\mathbb{R}}^N\right)$. In the above, ${(-{\Delta }_p)}^s$ is the fractional p-Laplacian, $s\in (0,1)$, $p>1$, $0<\alpha <ps<N$, $p_s^{\ast }=\frac{Np}{N-ps}$ is the critical fractional Sobolev exponent and $p_s^{\ast }\left(\alpha \right)=\frac{p(N-\alpha )}{N-ps}$ is the critical Hardy–Sobolev exponent, ${\dot{W}}^{s,p}\left({\mathbb{R}}^N\right)$ denotes the completion of $C_0^{\infty }\left({\mathbb{R}}^N\right)$ with respect to Gagliardo norm ${\left[u\right]}_{s,p}^p≔{\int }_{{\mathbb{R}}^N}{\int }_{{\mathbb{R}}^N}\frac{{|u\left(x\right)-u\left(y\right)|}^p}{{|x-y|}^{N+ps}}dxdy.$Our method is based on the existence of extremals of some fractional Hardy–Sobolev type inequalities, and coupled with some intricate estimates for the nonlocal (s,p)-gradient. Moreover, we also establish the existence of a nontrivial solution to an elliptic system which involves fractional p-Laplacian and critical Hardy–Sobolev exponents in ${\mathbb{R}}^N$.

使用Ambrosetti和Rabinowitz的Mountain-Pass定理，我们证明了以下分数 $p$-具有双临界非线性的拉普拉斯方程 ${（-{\Delta }_p）}^sü={\left|ü\right|}^{p_s^{\ast }-2}ü+\frac{{\left|ü\right|}^{p_s^{\ast }（\alpha ）-2}ü}{{\left|X\right|}^{\alpha }}\mathrm{一世}ñ{\mathbb{[R}}^{ñ}，$承认班上的正面解决方案 ${\dot{\mathrm{w\; ^}}}^{s，p}（{\mathbb{[R}}^{ñ}）$。在上面，${（-{\Delta }_p）}^s$ 是分数p-Laplacian， $s\in （0，\mathrm{1个}）$， $p>\mathrm{1个}$， $0<\alpha <ps<ñ$， $p_s^{\ast }=\frac{ñp}{ñ-ps}$ 是Sobolev的临界分数指数，并且 $p_s^{\ast }（\alpha ）=\frac{p（ñ-\alpha ）}{ñ-ps}$ 是关键的Hardy–Sobolev指数， ${\dot{\mathrm{w\; ^}}}^{s，p}（{\mathbb{[R}}^{ñ}）$ 表示完成 $C_0^{\infty }（{\mathbb{[R}}^{ñ}）$ 关于加利亚多准则 ${\left[ü\right]}_{s，p}^p≔{\int }_{{\mathbb{[R}}^{ñ}}{\int }_{{\mathbb{[R}}^{ñ}}\frac{{|ü（X）-ü（ÿ）|}^p}{{|X-ÿ|}^{ñ+ps}}dXdÿ。$我们的方法基于一些Hardy–Sobolev型不等式的极值的存在，并结合了一些非局部（s，p）梯度的复杂估计。此外，我们还建立了椭圆系统的非平凡解的存在性，该系统涉及分数p-Laplacian和临界Hardy-Sobolev指数。${\mathbb{[R}}^{ñ}$。