In this paper, we show the existence and multiplicity of positive solutions of the fractional Kirchhoff system$$\begin{aligned} {\left\{ \begin{array}{ll}\mathcal {L}_{M}(u) = {\lambda }f(x)|u|^{q-2}{u} + \frac{2{\alpha }}{{\alpha }+{\beta }} |u|^{\alpha -2} \,u|v|^{\beta} &{}\quad \mathrm{in}\, \Omega ,\\ \mathcal {L}_M(v) = {\mu }g(x)|v|^{q-2}v + \frac{2{\beta }}{{\alpha }+{\beta }}|u|^{\alpha} \,|v|^{\beta -2}v &{}\quad \mathrm{in}\, \Omega ,\\ \quad \;\;\; u = v = 0 &{}\quad \mathrm{on}\, {\partial }{\Omega }, \end{array}\right. } \end{aligned}$$where \(\mathcal{L}_{M}(u) = M \big(\int_{\Omega} |(-{\Delta})^\frac{s}{2}u|^{2}dx\big) (-\Delta)^{s}u\) is a double non-local operator due to a Kirchhoff term \(M(t) = a + bt\) with a, b > 0 and the fractional Laplacian \((-\Delta)^{s}, s \in (0,1)\). We consider that \(\Omega\) is an open and bounded domain in \(\mathbb{R}^{N}, 2s < N \leq 4s\) with smooth boundary, f, g are sign-changing continuous functions, \(\lambda,\mu > 0\) are real parameters, \(1 < q < 2, \alpha,\beta \geq 2\) and \(\alpha + \beta = 2_{s}^{*} = 2N/(N-2s)\) is a fractional critical exponent. Using the idea of Nehari manifold technique and a compactness result based on the classical idea of the Brezis–Lieb lemma, we prove the existence of at least two positive solutions for \((\lambda,\mu)\) lying in a suitable subset of \(\mathbb{R}^{2}_{+}\).

中文翻译：

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具有临界非线性分数阶Kirchhoff系统的Nehari流形

在本文中，我们展示了分数基希霍夫系统正解的存在性和多重性$$ \ begin {aligned} {\ left \ {\ begin {array} {ll} \ mathcal {L} _ {M}（u） = {\ lambda} f（x）| u | ^ {q-2} {u} + \ frac {2 {\ alpha}} {{\ alpha} + {\ beta}} | u | ^ {\ alpha- 2} \，u | v | ^ {\ beta}＆{} \ quad \ mathrm {in} \，\ Omega，\\ \ mathcal {L} _M（v）= {\ mu} g（x）| v | ^ {q-2} v + \ frac {2 {\ beta}} {{\ alpha} + {\ beta}} | u | ^ {\ alpha} \，| v | ^ {\ beta -2} v ＆{} \ quad \ mathrm {in} \，\ Omega，\\ \ quad \; \; \; u = v = 0＆{} \ quad \ mathrm {on} \，{\ partial} {\ Omega}，\ end {array} \ right。} \ end {aligned} $$其中\（\ mathcal {L} _ {M}（u）= M \ big（\ int _ {\ Omega} |（-{\ Delta}）^ \ frac {s} {2 } u | ^ {2} dx \ big）（-\ Delta）^ {s} u \）是双重非本地运算符，这是因为Kirchhoff项\（M（t）= a + bt \）带有a， b> 0和小数拉普拉斯算子\（（-\ Delta）^ {s}，s \ in（0,1）\）。我们认为\（\ Omega \）是\（\ mathbb {R} ^ {N}，2s <N \ leq 4s \）中具有光滑边界的开放有界域，f，g是符号改变的连续函数，\（\ lambda，\ mu> 0 \）是实际参数\（1 <q <2，\ alpha，\ beta \ geq 2 \）和\（\ alpha + \ beta = 2_ {s} ^ {*} = 2N /（N-2s）\）是分数临界指数。利用Nehari流形技术的思想和基于Brezis-Lieb引理经典思想的紧致性结果，我们证明了在适当子集中\（（\ lambda，\ mu）\）至少存在两个正解的\（\ mathbb {R} ^ {2} _ {+} \）。