We are concerned with the following elliptic equations with variable exponents:M([u]s,p(⋅,⋅))Lu(x)+V(x)|u|p(x)−2u=λρ(x)|u|r(x)−2u+h(x,u)in RN,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ M \\bigl([u]_{s,p(\\cdot,\\cdot)} \\bigr)\\mathcal{L}u(x) +\\mathcal {V}(x) \\vert u \\vert ^{p(x)-2}u =\\lambda\\rho(x) \\vert u \\vert ^{r(x)-2}u + h(x,u) \\quad \\text{in } \\mathbb {R}^{N}, $$\\end{document} where [u]_{s,p(cdot,cdot)}:=int_{mathbb {R}^{N}}int_{mathbb {R}^{N}} frac{|u(x)-u(y)|^{p(x,y)}}{p(x,y)|x-y|^{N+sp(x,y)}} ,dx ,dy, the operator mathcal{L} is the fractional p(cdot)-Laplacian, p, r: {mathbb {R}^{N}} to(1,infty) are continuous functions, M in C(mathbb {R}^{+}) is a Kirchhoff-type function, the potential function mathcal {V}:mathbb {R}^{N} to(0,infty) is continuous, and h:mathbb {R}^{N}timesmathbb {R} tomathbb {R} satisfies a Carathéodory condition. Under suitable assumptions on h, the purpose of this paper is to show the existence of at least two non-trivial distinct solutions for the problem above for the case of a combined effect of concave–convex nonlinearities. To do this, we use the mountain pass theorem and variant of the Ekeland variational principle as the main tools.