## Rocky Mountain Journal of Mathematics

### Nontrivial solutions for Kirchhoff-type problems involving the $p(x)$-Laplace operator

Abdeljabbar Ghanmi

#### Abstract

In this article, we study the existence of nontrivial solutions for the following $p(x)$ Kirchhoff-type problem $$\begin{cases}\smash {\!-M\big (\textstyle \int _{\Omega }A(x,\nabla u)\,dx\big ){div}(a(x,\nabla u))} =\lambda h(x)\frac{\partial F}{\partial u} (x,u), \quad \mbox {in } \Omega \\ u=0, \quad \mbox {on } \partial \Omega , \end{cases}$$ where $\Omega \subset \mathbb {R}^{n}$, $n\geq 3$, is a smooth bounded domain, $\lambda >0$, $h\in C(\Omega )$, $F:\overline {\Omega }\times \mathbb {R}\rightarrow \mathbb {R}$ is continuously differentiable and $a, A:\Omega \times \mathbb {R}^{n}\rightarrow \mathbb {R}^{n}$ are continuous. The proof is based on variational arguments and the theory of variable exponent Sobolev spaces.

#### Article information

Source
Rocky Mountain J. Math., Volume 48, Number 4 (2018), 1145-1158.

Dates
First available in Project Euclid: 30 September 2018

https://projecteuclid.org/euclid.rmjm/1538272826

Digital Object Identifier
doi:10.1216/RMJ-2018-48-4-1145

Mathematical Reviews number (MathSciNet)
MR3859751

Zentralblatt MATH identifier
06958772

#### Citation

Ghanmi, Abdeljabbar. Nontrivial solutions for Kirchhoff-type problems involving the $p(x)$-Laplace operator. Rocky Mountain J. Math. 48 (2018), no. 4, 1145--1158. doi:10.1216/RMJ-2018-48-4-1145. https://projecteuclid.org/euclid.rmjm/1538272826

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