On a class of fractional p(x) -Kirchhoff type problems

Abstract

This paper is concerned with a class of fractional $p\left(x\right)$-Kirchhoff type problems with Dirichlet boundary data of the following form $\left({\mathcal{P}}_M^s\right)\left\{\begin{array}{l}M\left({\int }_Q\frac{1}{p(x,y)}\frac{|u\left(x\right)-u\left(y\right){|}^{p(x,y)}}{|x-y{|}^{N+sp(x,y)}}\mathrm{d}x\mathrm{d}y\right){\left(-{\mathrm{\Delta }}_{p\left(x\right)}\right)}^{s}u\left(x\right)\\ =\lambda |u\left(x\right){|}^{r\left(x\right)-2}u\left(x\right)\text{in }\mathrm{\Omega },\\ u=0\text{in }{\mathbb{R}}^N\setminus \mathrm{\Omega }.\end{array}\right.$ By means of mountain pass theorem of Ambrosetti and Rabinowitz, direct variational approach and Ekeland's variational principle, we investigate the existence of nontrivial weak solutions for the above problem in different cases of the competition between the growth rates of functions p and r involved in problem $\left({\mathcal{P}}_M^s\right)$, this fact is essential in describing the set of eigenvalues of this problem.

COMMUNICATED BY:

• A. Pankov

Keywords:

• Kirchhoff type problems
• fractional $p\left(x\right)$-Laplacian operator
• montain pass theorem
• direct variatinal method
• Ekeland's variational principle

2010 Mathematics Subject Classifications:

• 35R11
• 35D30
• 35J35
• 35S15

Acknowledgments

We wish to express our gratitude to the referee for reading this paper carefully and making constructive comments and remarks.

Disclosure statement

No potential conflict of interest was reported by the authors.