In this article, we study the following fractional (p, q)-Laplacian equations involving the critical Sobolev exponent: $$({P_{\mu ,\lambda }})\left\{ {\begin{array}{*{20}{l}} {( - \Delta )_p^{{s_1}}u + ( - \Delta )_q^{{s_2}}u = \mu |u{|^{q - 2}}u + \lambda |u{|^{p - 2}}u + |u{|^{p_{{s_1}}^* - 2}}u,}&{\text{in}\;\Omega ,} \\ {u = 0,}&{\text{in}\;{\mathbb{R}^N}\backslash \Omega ,} \end{array}} \right.$$ where Ω ⊂ ℝN is a smooth and bounded domain, λ, μ > 0, 0 < S2 < s1 < 1, $$1 < q < p < {\textstyle{N \over {{s_1}}}}.$$ We establish the existence of a non-negative nontrivial weak solution to (Pμ,λ) by using the Mountain Pass Theorem. The lack of compactness associated with problems involving critical Sobolev exponents is overcome by working with certain asymptotic estimates for minimizers.

中文翻译：

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涉及临界 Sobolev 指数的分数 (p, q)-拉普拉斯问题的解的存在性

在本文中，我们研究以下涉及临界 Sobolev 指数的分数 (p, q)-拉普拉斯方程： $$({P_{\mu ,\lambda }})\left\{ {\begin{array}{*{ 20}{l}} {( - \Delta )_p^{{s_1}}u + ( - \Delta )_q^{{s_2}}u = \mu |u{|^{q - 2}}u + \lambda |u{|^{p - 2}}u + |u{|^{p_{{s_1}}^* - 2}}u,}&{\text{in}\;\Omega ,} \ \ {u = 0,}&{\text{in}\;{\mathbb{R}^N}\backslash \Omega ,} \end{array}} \right.$$ 其中 Ω ⊂ ℝN 是一个平滑且bounded domain, λ, μ > 0, 0 < S2 < s1 < 1, $$1 < q < p < {\textstyle{N \over {{s_1}}}}.$$ 我们建立了一个非负的存在使用 Mountain Pass 定理对 (Pμ,λ) 的非平凡弱解。与涉及关键 Sobolev 指数的问题相关的缺乏紧凑性可以通过使用某些极小值的渐近估计来克服。