Multiplicity of solutions to $(p,q)$-Kirchhoff equations with weight functions via local Palais-Smale condition

In this paper, we study the following $(p,q)$-Kirchhoff equation with weight functions:

$$\{\begin{array}{rl}& \sum _{k\in \{p,q\}}\left(-\left(1+{\displaystyle {\int }_{{\mathbb{R}}^N}|\mathrm{\nabla }u{|}^{k}dx}\right){\mathrm{\Delta }}_{k}u+|u{|}^{k-2}u\right)=\lambda h(x)|u{|}^{r-2}u+g(x)|u{|}^{s-2}u,\\ & u\in W^{1,p}({\mathbb{R}}^N)\cap W^{1,q}({\mathbb{R}}^N),\end{array}$$

where $1<q\le p$, $\{r,s\}\subset (2p,p\ast )$, and $p<N<2p$. By applying a critical point theorem following Perera (J. Anal. Math., 2025), there exists ${\mathrm{\Lambda }}_m\ge 0$ such that the equation has at least $m\in \mathbb{N}$ pairs of nontrivial solutions for every $\lambda >{\mathrm{\Lambda }}_m$. Particularly, we need only prove that the corresponding energy functional satisfies the local Palais-Smale condition, and an explicit expression for ${\mathrm{\Lambda }}_m$ is given, which generalizes some results in the existing literature and provides a new perspective for searching for multiple solutions of Kirchhoff-type equations.