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Volume 36 • Number 3 • 2013

• Existence and Multiplicity of Solutions for $p(x)$ -Kirchhoff-Type Problem in $\mathbf{R}^N$
Mei-Chun Wei and Chun-Lei Tang

Abstract.
In this paper we study the

$p(x)$ -Kirchhoff-type problem

$\begin{equation*} \begin{cases} -m\left(\int_{\text{R}^N}\frac{1}{p(x)}|\nabla u|^{p(x)} dx\right) \text{div}(|\nabla u|^{p(x)-2}\nabla u)+|u|^{p(x)-2}u={f(x,u)}\quad \text{T}{in}\ \text{R}^N\\ u\in W^{1,p(x)}(\text{R}^N). \end{cases} \end{equation*}$ We first establish the compact imbedding

$W^{1,p(x)}(\text{R}^N)\hookrightarrow L_{b(x)}^{q(x)}(\text{R}^N)$ , where

$L_{b(x)}^{p(x)}( \text{R}^N)=$

$\{u$ is measurable on

$\text{R}^N$ :

$\int_{\text{R}^N}b(x)|u|^{p(x)}dx<\infty\}$ . Based on it, the existence and multiplicity of solutions for the problem are obtained by variational methods.

2010 Mathematics Subject Classification: 35J35, 35J60, 47J30, 58E05

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