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

Ghasem A. Afrouzi; Maryam Mirzapour; Nguyen Thanh Chung

doi:10.4171/zaa/1512

JournalszaaVol. 33, No. 3pp. 289–303

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

Ghasem A. Afrouzi
University of Mazandaran, Babolsar, Iran
Maryam Mirzapour
University of Mazandaran, Babolsar, Iran
Nguyen Thanh Chung
Quang Binh University, Vietnam

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Abstract

This paper is concerned with the existence and multiplicity of weak solutions for a $p (x)$ -Kirchhoff type problem of the following form

⎩ ⎨ ⎧ M (\int_{Ω} \frac{1}{p ( x )} ∣Δ u ∣^{p (x)} d x) Δ (∣Δ u ∣^{p (x) - 2} Δ u) u = f (x, u) = Δ u = 0 in Ω on \partial Ω,

by using the mountain pass theorem of Ambrosetti and Rabinowitz and Ekeland's variational principle in two cases when the Carathéodory function $f (x, u)$ having special structure.

Cite this article

Ghasem A. Afrouzi, Maryam Mirzapour, Nguyen Thanh Chung, Existence and Multiplicity of Solutions for Kirchhoff Type Problems Involving $p (x)$ -Biharmonic Operators. Z. Anal. Anwend. 33 (2014), no. 3, pp. 289–303

DOI 10.4171/ZAA/1512