The paper deals with the existence and multiplicity of nontrivial weak solutions for the p(x)-Kirchhofftype problem

$$ {\displaystyle \begin{array}{c}-M\left(\underset{\Omega}{\int}\frac{1}{p(x)}{\left|\Delta u\right|}^{p(x)} dx\right){\Delta}_{p(x)}^2u=f\left(x,u\right)\kern0.6em \mathrm{in}\kern0.48em \Omega, \\ {}u=\Delta u=0\kern0.48em \mathrm{on}\kern0.48em \mathrm{\partial \Omega }.\end{array}} $$

By using the variational approach and the Krasnosel’skii genus theory, we prove the existence and multiplicity of solutions for the p(x)-Kirchhoff-type equation.

中文翻译：

---

基于类理论的涉及p（X）-双调和算子的p（X）-Kirchhoff-型方程

本文讨论了p（x）-Kirchhoff型问题的非平凡弱解的存在性和多重性。

$$ {\ displaystyle \ begin {array} {c} -M \ left（\ underset {\ Omega} {\ int} \ frac {1} {p {x）} {\ left | \ Delta u \ right |} ^ {p（x）} dx \ right）{\ Delta} _ {p（x）} ^ 2u = f \ left（x，u \ right）\ kern0.6em \ mathrm {in} \ kern0.48em \ Omega ，\\ {} u = \ Delta u = 0 \ kern0.48em \ mathrm {on} \ kern0.48em \ mathrm {\ partial \ Omega}。\ end {array}} $$

通过使用变分方法和Krasnosel'skii类理论，我们证明了p（x）-Kirchhoff型方程的解的存在性和多重性。