Rendiconti del Seminario Matematico della Università di Padova


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Volume 136, 2016, pp. 95–109
DOI: 10.4171/RSMUP/136-8

Published online: 2016-12-22

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

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 to $p(x)$-Kirchhoff type problem of the following form $$-M \big{(} \int_{\Omega}\frac{1}{p(x)}|\nabla u|^{p(x)}dx\big {)}\mathrm{div}(|\nabla u|^{p(x)-2}\nabla u) =f(x,u) \quad \text{in } \Omega, $$ $$u=0 \quad \mathrm {on} \: \partial \Omega.$$ 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.

Keywords: Kirchhoff type problems; $p(x)$-Kirchhoff type, boundary value problem, mountain pass theorem, dual fountain theorem

Afrouzi Ghasem, Mirzapour Maryam, Chung Nguyen Thanh: Existence and multiplicity of solutions for a $p(x)$-Kirchhoff type equation. Rend. Sem. Mat. Univ. Padova 136 (2016), 95-109. doi: 10.4171/RSMUP/136-8