In this paper, we consider the doubly critical coupled systems involving fractional Laplacian in ${ℝ}^n$ with partial singular weight:

$$\left\{\begin{array}{l}{(-\mathrm{\Delta })}^{s}u-{\gamma }_1\frac{u}{|x^{\prime }{|}^{2s}}=\frac{{|u|}^{2_s^{\ast }(\beta )-2}u}{|x^{\prime }{|}^{\beta }}+\frac{{\eta }_1}{2_s^{\ast }(\alpha )}\frac{{|u|}^{{\eta }_1-2}u{|v|}^{{\eta }_2}}{|x^{\prime }{|}^{\alpha }},\\ {(-\mathrm{\Delta })}^{s}v-{\gamma }_2\frac{v}{|x^{\prime }{|}^{2s}}=\frac{{|v|}^{2_s^{\ast }(\beta )-2}v}{|x^{\prime }{|}^{\beta }}+\frac{{\eta }_2}{2_s^{\ast }(\alpha )}\frac{{|v|}^{{\eta }_2-2}v{|u|}^{{\eta }_1}}{|x^{\prime }{|}^{\alpha }},\end{array}\right.$$ (0.1)

where s ∈ (0, 1), 0 ≤ α, β < 2s < n, 0 < m < n, $x=(x^{\prime },x^{\prime \prime })\in {ℝ}^m\times {ℝ}^{n-m}$, η₁, η₂ > 1, ${\eta }_1+{\eta }_2=2_s^{\ast }(\alpha ):=2(n-\alpha )/n-2s$, γ₁, γ₂ < γ_H, and ${\gamma }_H={\gamma }_H(n,m,s)>0$ is some explicit constant. By establishing new embedding results involving partially weighted Morrey norms in the product space ${\dot{H}}^s({ℝ}^n)\times {\dot{H}}^s({ℝ}^n)$, we provide sufficient conditions under which a weak nontrivial solution of (0.1) exists via variational methods. We also extend these results to p-Laplacian systems especially.

中文翻译：

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关于具有偏奇异权的分数拉普拉斯算子的双临界耦合系统

在本文中，我们考虑涉及分数拉普拉斯算子的双临界耦合系统 ${ℝ}^n$ 具有部分奇异权重：

$$\left\{\begin{array}{l}{(-\mathrm{\Delta })}^{秒}你-{\gamma }_1\frac{你}{|X^{\prime }{|}^{2秒}}=\frac{{|你|}^{2_{秒}^{＊}(\beta )-2}你}{|X^{\prime }{|}^{\beta }}+\frac{{\eta }_1}{2_{秒}^{＊}(\alpha )}\frac{{|你|}^{{\eta }_1-2}你{|v|}^{{\eta }_2}}{|X^{\prime }{|}^{\alpha }},\\ {(-\mathrm{\Delta })}^{秒}v-{\gamma }_2\frac{v}{|X^{\prime }{|}^{2秒}}=\frac{{|v|}^{2_{秒}^{＊}(\beta )-2}v}{|X^{\prime }{|}^{\beta }}+\frac{{\eta }_2}{2_{秒}^{＊}(\alpha )}\frac{{|v|}^{{\eta }_2-2}v{|你|}^{{\eta }_1}}{|X^{\prime }{|}^{\alpha }},\end{array}\right.$$ (0.1)

其中s ∈ (0, 1), 0 ≤ α ,  β < 2 s < n , 0 < m < n ,$X=(X^{\prime },X^{\prime \prime })\in {ℝ}^{米}\times {ℝ}^{n-米}$, η ₁ ,  η ₂ > 1,${\eta }_1+{\eta }_2=2_{秒}^{＊}(\alpha )：=2(n-\alpha )/n-2秒$, γ ₁ ,  γ ₂ < γ _H , 和${\gamma }_H={\gamma }_H(n,米,秒)>0$是一些显式常数。通过在产品空间中建立涉及部分加权莫雷范数的新嵌入结果${\dot{H}}^{秒}({ℝ}^n)\times {\dot{H}}^{秒}({ℝ}^n)$，我们提供了通过变分方法存在（0.1）的弱非平凡解的充分条件。我们还将这些结果特别扩展到p- Laplacian 系统。