In this paper, we are concerned with the existence of least energy sign-changing solutions for the following fractional Kirchhoff problem with logarithmic and critical nonlinearity:$$\begin{aligned} \left\{ \begin{array}{ll} \left( a+b[u]_{s,p}^p\right) (-\Delta )^s_pu = \lambda |u|^{q-2}u\ln |u|^2 + |u|^{ p_s^{*}-2 }u &{}\quad \text {in } \Omega , \\ u=0 &{}\quad \text {in } {\mathbb {R}}^N{\setminus } \Omega , \end{array}\right. \end{aligned}$$where \(N >sp\) with \(s \in (0, 1)\), \(p>1\), and$$\begin{aligned}{}[u]_{s,p}^p =\iint _{{\mathbb {R}}^{2N}}\frac{|u(x)-u(y)|^p}{|x-y|^{N+ps}}dxdy, \end{aligned}$$\(p_s^*=Np/(N-ps)\) is the fractional critical Sobolev exponent, \(\Omega \subset {\mathbb {R}}^N\)\((N\ge 3)\) is a bounded domain with Lipschitz boundary and \(\lambda \) is a positive parameter. By using constraint variational methods, topological degree theory and quantitative deformation arguments, we prove that the above problem has one least energy sign-changing solution \(u_b\). Moreover, for any \(\lambda > 0\), we show that the energy of \(u_b\) is strictly larger than two times the ground state energy. Finally, we consider b as a parameter and study the convergence property of the least energy sign-changing solution as \(b \rightarrow 0\).

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具有对数非线性的关键Kirchhoff问题的最小能量节点解

在本文中，我们关注以下具有对数和临界非线性的分数阶Kirchhoff问题的最小能量符号转换解的存在：$$ \ begin {aligned} \ left \ {\ begin {array} {ll} \ left （a + b [u] _ {s，p} ^ p \ right）（-\ Delta）^ s_pu = \ lambda | u | ^ {q-2} u \ ln | u | ^ 2 + | u | ^ {p_s ^ {**-2）} u＆{} \ quad \ text {in} \ Omega，\\ u = 0＆{} \ quad \ text {in} {\ mathbb {R}} ^ N {\ setminus } \ Omega，\ end {array} \ right。\ end {aligned} $$其中\（N> sp \）与\（s \ in（0，1）\），\（p> 1 \）和$$ \ begin {aligned} {} [u] _ {s，p} ^ p = \ iint _ {{\ mathbb {R}} ^ {2N}} \ frac {| u（x）-u（y）| ^ p} {| xy | ^ {N + ps}} dxdy，\ end {aligned} $$ \（p_s ^ * = Np /（N-ps）\）是分数临界Sobolev指数，\（\ Omega \ subset {\ mathbb {R}} ^ N \）\（（N \ ge 3）\）是具有Lipschitz边界的有界域，而\（\ lambda \）是一个正参数。通过使用约束变分方法，拓扑度理论和定量变形参数，我们证明上述问题具有一个最小的能量符号改变解\（u_b \）。此外，对于任何\（\ lambda> 0 \），我们证明\（u_b \）的能量严格大于基态能量的两倍。最后，我们将b作为参数，并研究最小能量符号改变解的收敛性\（b \ rightarrow 0 \）。