Existence and Multiplicity of Solutions for $p(x)$-Kirchhoff-Type Problem in $\mathbf{R}^N$

Mei-Chun Wei ; Chun-Lei Tang

Bulletin of the Malaysian Mathematical Society, Tome 36 (2013) no. 3

Résumé

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.

Détail

Classification : 35J35, 35J60, 47J30, 58E05