The Nehari manifold for anisotropic Kirchhoff problems involving variable singular exponent and critical terms

In this paper, we investigate multiplicity results for the following anisotropic Kirchhoff-type problems with a singular term and a critical nonlinearity that changes sign \(\begin{aligned} {\left\{ \begin{array}{ll} -\sum \limits _{i=1}^{N} M\left( \int _{\Omega }|\partial _i u|^{p_i}dx\right) \partial _{i} \left( {| \partial _{i} u \vert }^{p_{i}-2} \partial _{i} u \right) = \frac{ f_1(x)}{u^{\beta (x)}}+ \lambda f_2(x) u^{\bar{p}^* -1} & \qquad \text {in }\;\;\Omega ,\\ u>0 & \qquad \text {in } \;\;\Omega , \\ u=0 & \qquad \text {on } \;\;\partial \Omega , \\ \end{array}\right. } \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{cc}-\sum _{i=1}^{N}M\left({\int }_{\mathrm{\Omega }},{|{\partial }_{i}u|}^{p_i},d,x\right){\partial }_i\left({|{\partial }_{i}u|}^{p_i-2},{\partial }_i,u\right)=\frac{f_1(x)}{u^{\beta (x)}}+\lambda f_2(x)u^{{\overline{p}}^{\ast }-1}\hfill & \text{in}\mathrm{\Omega },\hfill \\ u>0\hfill & \text{in}\mathrm{\Omega },\hfill \\ u=0\hfill & \text{on}\partial \mathrm{\Omega },\hfill \end{array}\right)\end{array}$$ where \(\;\Omega \;\) $\mathrm{\Omega }$ is a bounded regular domain in \(\;{\mathbb {R}}^{N} \) ${\mathbb{R}}^N$ , \(N>\bar{p} \) $N>\overline{p}$ and the critical Sobolev exponent, denoted as \(\bar{p}^*\) ${\overline{p}}^{\ast }$ , is defined as \(\bar{p}^*=N\bar{p}/(N-\bar{p})\) ${\overline{p}}^{\ast }=N\overline{p}/(N-\overline{p})$ , where \(\bar{p}=N/\sum _{i=1}^{N} \frac{1}{p_{i}}\) $\overline{p}=N/{\sum }_{i=1}^N\frac{1}{p_i}$ . Additionally, we have the parameter \(\lambda >0\) $\lambda >0$ and the exponent variable \(0<\beta (x)<1\) $0<\beta (x)<1$ . Here, \(f_1\) $f_1$ is a positive function, while \(f_2\) $f_2$ is a function that changes sign, and M models a Kirchhoff coefficient. We establish the existence of at least two weak solutions that have different energy sign. Our approach relies on the fibering method in the form of the Nehari manifold.