Rendiconti del Seminario Matematico della Università di Padova


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

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) Department of Mathematics, University of Mazandaran, Babolsar, Iran
(2) Department of Mathematics, University of Mazandaran, Babolsar, Iran
(3) Department of Mathematics, Quang Binh University, 312 Ly Thuong Kiet, Dong Hoi, Quang Binh, Viet Nam

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