Existence and multiplicity of solutions for a $p(x)$-Kirchhoff type problem via variational techniques

Mokhtari, A. ; Moussaoui, T. ; O’Regan, D.

Archivum mathematicum, Tome 51 (2015) no. 3, pp. 163-173.

Résumé

This paper discusses the existence and multiplicity of solutions for a class of $p(x)$-Kirchhoff type problems with Dirichlet boundary data of the following form \[ {\left\rbrace \begin{array}{ll} -\Big (a+b\int _{\Omega }\frac{1}{p(x)}|\nabla u|^{p(x)}\; dx\Big )\textrm{div}\big (|\nabla u|^{p(x)-2 } \nabla u\big )= f(x,u)\,, in \quad \Omega \\[6pt] u=0 on \quad \partial \Omega\,, \end{array}\right.} \] where $\Omega $ is a smooth open subset of $\mathbb{R}^N$ and $p\in C(\overline{\Omega })$ with $N $, $a$, $b$ are positive constants and $f\colon \overline{\Omega }\times \mathbb{R}\rightarrow \mathbb{R}$ is a continuous function. The proof is based on critical point theory and variable exponent Sobolev space theory.

Détail

DOI : 10.5817/AM2015-3-163

Classification : 34B27, 35B05, 35J60
Keywords: existence results; genus theory; nonlocal problems Kirchhoff equation; critical point theory