## Existence and multiplicity results for a new $p(x)$-Kirchhoff problem

22 Aug 2019  ·  Hamdani M. K., Harrabi A., Mtiri F., Repovš D. D. ·

We study the existence and multiplicity results for the following nonlocal $p(x)$-Kirchhoff problem: $$\label{10} \begin{cases} -\left(a-b\int_\Omega\frac{1}{p(x)}| \nabla u| ^{p(x)}dx\right)div(|\nabla u| ^{p(x)-2}\nabla u)=\lambda |u| ^{p(x)-2}u+g(x,u) \mbox{ in } \Omega, \\ u=0,\mbox{ on } \partial\Omega, \end{cases}$$ where $a\geq b > 0$ are constants, $\Omega\subset \mathbb{R}^N$ is a bounded smooth domain, $p\in C(\overline{\Omega})$ with $N>p(x)>1$, $\lambda$ is a real parameter and $g$ is a continuous function. The analysis developed in this paper proposes an approach based on the idea of considering a new nonlocal term which presents interesting difficulties...

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Analysis of PDEs