# Article

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Keywords:
existence results; genus theory; nonlocal problems Kirchhoff equation; critical point theory
Summary:
This paper discusses the existence and multiplicity of solutions for a class of $p(x)$-Kirchhoff type problems with Dirichlet boundary data of the following form ${\left\rbrace \begin{array}{ll} -\Big (a+b\int _{\Omega }\frac{1}{p(x)}|\nabla u|^{p(x)}\; dx\Big )\textrm{div}\big (|\nabla u|^{p(x)-2 } \nabla u\big )= f(x,u)\,, & in \quad \Omega \\[6pt] u=0 & on \quad \partial \Omega\,, \end{array}\right.}$ where $\Omega$ is a smooth open subset of $\mathbb{R}^N$ and $p\in C(\overline{\Omega })$ with $N <p^-= \inf _{x\in \Omega } p(x)\le p^+= \sup _{x\in \Omega } p(x)<+\infty$, $a$, $b$ are positive constants and $f\colon \overline{\Omega }\times \mathbb{R}\rightarrow \mathbb{R}$ is a continuous function. The proof is based on critical point theory and variable exponent Sobolev space theory.
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