Existence and multiplicity of solutions for fractional p(x,.)-Kirchhoff-type problems in ℝ^N

Abstract

The purpose of this paper is to investigate the existence and multiplicity of solutions for the following class of fractional $p(x,.)$-Kirchhoff-type problems in ${\mathbb{R}}^N$ $\begin{array}{r}\left({\mathcal{P}}_M^s\right)\left\{\begin{array}{l}M\left({\int }_{{\mathbb{R}}^N\times {\mathbb{R}}^N}\frac{1}{p(x,y)}\frac{|u\left(x\right)-u\left(y\right){|}^{p(x,y)}}{|x-y{|}^{N+sp(x,y)}}\mathrm{d}x\mathrm{d}y\right){\left(-{\mathrm{\Delta }}_{p(x,.)}\right)}^{s}u\left(x\right)\\ +|u{|}^{\overline{p}\left(x\right)-2}u=f(x,u)\mathrm{i}\mathrm{n}{\mathbb{R}}^N,\\ u\in W^{s,p(x,y)}\left({\mathbb{R}}^N\right).\end{array}\right.\end{array}$ We first establish the compact embedding of the space $W^{s,p(x,y)}\left({\mathbb{R}}^N\right)$ into $L_{w\left(x\right)}^{q\left(x\right)}\left({\mathbb{R}}^N\right)$. Using this embedding result and some critical-points theorems, we obtain the existence and multiplicity results.

KEYWORDS:

• Fractional $p(x,.)$-Kirchhoff-type problems
• generalized fractional sobolev space
• compact embedding
• fountain theorem
• symmetric mountain pass theorem

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 35R11
• 35J35
• 35S15
• 46E35

Disclosure statement

No potential conflict of interest was reported by the authors.