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Zeitschrift für Analysis und ihre Anwendungen


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Volume 33, Issue 3, 2014, pp. 289–303
DOI: 10.4171/ZAA/1512

Published online: 2014-07-02

Existence and Multiplicity of Solutions for Kirchhoff Type Problems Involving $p(x)$-Biharmonic Operators

Ghasem A. Afrouzi[1], Maryam Mirzapour[2] and Nguyen Thanh Chung[3]

(1) University of Mazandaran, Babolsar, Iran
(2) University of Mazandaran, Babolsar, Iran
(3) Quang Binh University, Vietnam

This paper is concerned with the existence and multiplicity of weak solutions for a $p(x)$-Kirchhoff type problem of the following form \begin{equation*} \left\{ \begin{alignedat}{2} M\left(\int_{\Omega}\frac{1}{p(x)}|\Delta u|^{p(x)}\,dx\right)\Delta(|\Delta u|^{p(x)-2}\Delta u) &=f(x,u)& &\text{ in } \Omega \\ u&=\Delta u =0 &\quad &\textrm{ on } \partial\Omega, \end{alignedat}\right. \end{equation*} by using the mountain pass theorem of Ambrosetti and Rabinowitz and Ekeland's variational principle in two cases when the Carath\'{e}odory function $f(x,u)$ having special structure.

Keywords: $p(x)$-biharmonic operators, Kirchhoff type problems, mountain pass theorem, Ekeland's variational principle

Afrouzi Ghasem, Mirzapour Maryam, Chung Nguyen Thanh: Existence and Multiplicity of Solutions for Kirchhoff Type Problems Involving $p(x)$-Biharmonic Operators. Z. Anal. Anwend. 33 (2014), 289-303. doi: 10.4171/ZAA/1512