A degenerate Kirchhoff-type problem involving variable s(·)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s(\cdot )$$\end{document}-order fractional p(·)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p(\cdot )$$\end{document}-Laplacian with weights

This paper deals with a class of nonlocal variable s(.)-order fractional p(.)-Kirchhoff type equations: K∫R2N1p(x,y)|φ(x)-φ(y)|p(x,y)|x-y|N+s(x,y)p(x,y)dxdy(-Δ)p(·)s(·)φ(x)=f(x,φ)inΩ,φ=0onRN\Ω.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left\{ \begin{array}{ll} {\mathcal {K}}\left( \int _{{\mathbb {R}}^{2N}}\frac{1}{p(x,y)}\frac{|\varphi (x)-\varphi (y)|^{p(x,y)}}{|x-y|^{N+s(x,y){p(x,y)}}} \,dx\,dy\right) (-\Delta )^{s(\cdot )}_{p(\cdot )}\varphi (x) =f(x,\varphi ) \quad \text{ in } \Omega , \\ \varphi =0 \quad \text{ on } {\mathbb {R}}^N\backslash \Omega . \end{array} \right. \end{aligned}$$\end{document}Under some suitable conditions on the functions p,s,K\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p,s, {\mathcal {K}}$$\end{document} and f, the existence and multiplicity of nontrivial solutions for the above problem are obtained. Our results cover the degenerate case in the p(·)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p(\cdot )$$\end{document} fractional setting.