Existence and multiplicity of solutions for a $p(x)$-Kirchhoff type equation

Ghasem A. Afrouzi; Maryam Mirzapour; Nguyen Thanh Chung

doi:10.4171/rsmup/136-8

JournalsrsmupVol. 136pp. 95–109

Existence and multiplicity of solutions for a $p (x)$ -Kirchhoff type equation

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 to $p (x)$ -Kirchhoff type problem of the following form

{- M (\int_{Ω} \frac{1}{p ( x )} ∣\nabla u ∣^{p (x)} d x) div (∣\nabla u ∣^{p (x) - 2} \nabla u) = f (x, u) u = 0 in Ω, on \partial Ω.

By means of a direct variational approach and the theory of the variable exponent Sobolev spaces, we establish conditions ensuring the existence and multiplicity of solutions for the problem.

Cite this article

Ghasem A. Afrouzi, Maryam Mirzapour, Nguyen Thanh Chung, Existence and multiplicity of solutions for a $p (x)$ -Kirchhoff type equation. Rend. Sem. Mat. Univ. Padova 136 (2016), pp. 95–109

DOI 10.4171/RSMUP/136-8