In this paper, we are interested in existence and multiplicity of solutions to anisotropic elliptic equations of Kirchhoff-type given by $\begin{array}{ll}L\left(u\right)=\lambda {\left({\int }_{\mathrm{\Omega }}F(x,u)\right)}^{r}f(x,u)+|u{|}^{p^{\ast }-2}u& \text{ in }\mathrm{\Omega },\\ u=0& \text{ on }\mathrm{\partial }\mathrm{\Omega },\end{array}(P{)}_{\lambda }$ where $\begin{array}{rl}L\left(u\right)& :=-\sum _{i=1}^{N}M\left({\int }_{\mathrm{\Omega }}{\left|\frac{\mathrm{\partial }u}{\mathrm{\partial }{x}_i}\right|}^{p_i}\right)\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_i}\\ & \times \left({\left|\frac{\mathrm{\partial }u}{\mathrm{\partial }{x}_i}\right|}^{p_i-2}\frac{\mathrm{\partial }u}{\mathrm{\partial }{x}_i}\right),\end{array}$ Ω is a smooth bounded domain of ${\mathbb{R}}^N,$ $N>2,$ $\lambda >0$ and $r\ge 0$ are parameters, $1<p_1\le \cdots .\le p_N<p^{\ast },$ where $p^{\ast }=N\overline{p}/(N-\overline{p})$ is the critical exponent and $\overline{p}=N/\sum _{i=1}^N\frac{1}{p_i}$. The function $M:[0,+\mathrm{\infty })\to (-\mathrm{\infty },+\mathrm{\infty })$ is only continuous with $M\left(0\right)>0$ and can change its sign, $f:\mathrm{\Omega }\times \mathbb{R}\to \mathbb{R}$ is a function with subcritical growth and $F(x,t)={\int }_0^{t}f(x,s)ds.$ Under appropriate assumptions on f, applying an adequate truncation argument on M and the Concentration Compactness-Principle for the anisotropic operator [1 El Hamidi A, Rakotoson JM. Extremal functions for the anisotropic Sobolev inequalities. Ann I H Poincaré AN. 2007;24:741–756.[Crossref], [Web of Science ®] , [Google Scholar]], we obtain the existence of a nontrivial solution for $(P{)}_{\lambda }$ by the Mountain Pass Theorem [2 Rabinowitz PH. Minimax methods in critical point theory with applications to differential equations. Providence RI: American Mathematical Society; 1986. (CBMS Reg Conf Ser Math; 65).[Crossref] , [Google Scholar]] and multiplicity of solutions by Krasnoselskii's genus and the symmetric Mountain Pass Lemma [3 Kajikiya R. A critical-point theorem related to the symmetric mountain-pass lemma and its applications to elliptic equations. J Funct Anal. 2005;225(2):352–370.[Crossref], [Web of Science ®] , [Google Scholar]]. As a model case for M and f, we can consider $M\left(t\right)=a+bt$ or $M\left(t\right)=a-bt$ with $a>0,b\ge 0$ and $f(x,t)=a\left(x\right)|t{|}^{q-2}t$ with $a\in L^{\mathrm{\infty }}\left(\mathrm{\Omega }\right)$ and appropriate $q\in (1,p_{\ast }).$

在本文中，我们感兴趣的是基尔霍夫型各向异性椭圆方程解的存在性和多重性$\begin{array}{ll}\mathrm{大号}\left(你\right)=\lambda {\left({\int }_{\mathrm{\Omega }}F(X,你)\right)}^{r}F(X,你)+|你{|}^{p^{*}-2}你& \text{ 在 }\mathrm{\Omega },\\ 你=0& \text{ 在 }\mathrm{\partial }\mathrm{\Omega },\end{array}(磷{)}_{\lambda }$在哪里$\begin{array}{rl}\mathrm{大号}\left(你\right)& :=-\sum _{\mathrm{一世}=1}^{ñ}米\left({\int }_{\mathrm{\Omega }}{\left|\frac{\mathrm{\partial }你}{\mathrm{\partial }{X}_{\mathrm{一世}}}\right|}^{p_{\mathrm{一世}}}\right)\frac{\mathrm{\partial }}{\mathrm{\partial }{X}_{\mathrm{一世}}}\\ & \times \left({\left|\frac{\mathrm{\partial }你}{\mathrm{\partial }{X}_{\mathrm{一世}}}\right|}^{p_{\mathrm{一世}}-2}\frac{\mathrm{\partial }你}{\mathrm{\partial }{X}_{\mathrm{一世}}}\right),\end{array}$Ω 是一个光滑有界域${\mathbb{R}}^{ñ},$ $ñ>2,$ $\lambda >0$和$r\ge 0$是参数，$1<p_1\le \cdots .\le p_{ñ}<p^{*},$在哪里$p^{*}=ñ\overline{p}/(ñ-\overline{p})$是临界指数，并且$\overline{p}=ñ/\sum _{\mathrm{一世}=1}^{ñ}\frac{1}{p_{\mathrm{一世}}}$. 功能$米：[0,+\mathrm{\infty })\to (-\mathrm{\infty },+\mathrm{\infty })$只与$米\left(0\right)>0$并且可以改变它的符号，$F：\mathrm{\Omega }\times \mathbb{R}\to \mathbb{R}$是一个具有亚临界增长的函数，并且$F(X,吨)={\int }_0^{吨}F(X,s)ds.$在对f的适当假设下，对M应用适当的截断参数以及各向异性算子的浓度紧致原则 [ 1 El Hamidi A , Rakotoson JM. 各向异性 Sobolev 不等式的极值函数。安 IH 庞加莱 AN。2007 年；24：741 – 756。[Crossref], [Web of Science ®]  , [Google Scholar] ]，我们得到一个非平凡解的存在性$(磷{)}_{\lambda }$由山口定理 [ 2 拉比诺维茨PH. 临界点理论中的极小极大方法与微分方程的应用。普罗维登斯 RI：美国数学会；1986 年。（CBMS Reg Conf Ser 数学；65）。[Crossref]  ， [Google Scholar] ] 和 Krasnoselskii 属和对称 Mountain Pass 引理的解的多样性 [3 Kajikiya R. 一个与对称山口引理相关的临界点定理及其在椭圆方程中的应用。J功能肛门。2005 年；225(2): 352 – 370。[交叉引用]、[Web of Science ®]  、[谷歌学术] ]。作为M和f，我们可以考虑$米\left(吨\right)=\mathrm{一种}+b吨$要么$米\left(吨\right)=\mathrm{一种}-b吨$和$\mathrm{一种}>0,b\ge 0$和$F(X,吨)=\mathrm{一种}\left(X\right)|吨{|}^{q-2}吨$和$\mathrm{一种}\in {\mathrm{大号}}^{\mathrm{\infty }}\left(\mathrm{\Omega }\right)$并且适当$q\in (1,p_{*}).$