This paper deals with the study of combined effects of logarithmic and critical nonlinearities for the following class of fractional $p$-Kirchhoff equations: $\left\{\begin{array}{cc}M\left({\left[u\right]}_{s,p}^p\right){(-\Delta )}_p^{s}u=\lambda {\left|u\right|}^{q-2}uln{\left|u\right|}^2+{\left|u\right|}^{p_s^{\ast }-2}u\hfill & \text{in}\Omega ,\hfill \\ u=0\hfill & \text{in}{\mathbb{R}}^N\setminus \Omega ,\hfill \end{array}\right.$where $\Omega \subset {\mathbb{R}}^N$ is a bounded domain with Lipschitz boundary, $N>sp$ with $s\in (0,1)$, $p\ge 2$, $p_s^{\ast }=Np∕(N-ps)$ is the fractional critical Sobolev exponent, and $\lambda$ is a positive parameter. The main result establishes the existence of nontrivial solutions in the case of high perturbations of the logarithmic nonlinearity (large values of $\lambda$). The features of this paper are the following: (i) the presence of a logarithmic nonlinearity; (ii) the lack of compactness due to the critical term; (iii) our analysis includes the degenerate case, which corresponds to the Kirchhoff term $M$ vanishing at zero.

本文针对以下几类分数对数和临界非线性的组合效应进行研究 $p$-Kirchhoff方程： $\left\{\begin{array}{cc}\mathrm{中号}（{\left[ü\right]}_{s，p}^p）{（-\Delta ）}_p^sü=\lambda {\left|ü\right|}^{q-2}üln{\left|ü\right|}^2+{\left|ü\right|}^{p_s^{\ast }-2}ü\hfill & \text{在}\Omega ，\hfill \\ ü=0\hfill & \text{在}{\mathbb{[R}}^{ñ}\setminus \Omega ，\hfill \end{array}\right.$哪里 $\Omega \subset {\mathbb{[R}}^{ñ}$ 是具有Lipschitz边界的有界域， $ñ>sp$ 与 $s\in （0，\mathrm{1个}）$， $p\ge 2$， $p_s^{\ast }=ñp∕（ñ-ps）$ 是分数临界Sobolev指数，并且 $\lambda$是一个正参数。主要结果证明了在对数非线性的高扰动情况下（非正则项的大值）存在非平凡解的存在。$\lambda$）。本文的特点如下：（i）存在对数非线性；（ii）由于关键术语而缺乏紧凑性；（iii）我们的分析包括退化的情况，它对应于基尔霍夫项$\mathrm{中号}$ 消失在零。