A degenerate Kirchhoff-type problem involving variable $s(\cdot )$-order fractional $p(\cdot )$-Laplacian with weights

This paper deals with a class of nonlocal variable $s(\mathrm{.})$-order fractional $p(\mathrm{.})$-Kirchhoff type equations: \begin{eqnarray*} \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)(-Δ)^{s(\cdot)}_{p(\cdot)}\varphi(x) =f(x,\varphi) \quad \mbox{in }Ω, \ \varphi=0 \quad \mbox{on }$\mathbb{R}$^N\backslashΩ. \end{array} \right. \end{eqnarray*} Under some suitable conditions on the functions $p,s,\mathcal{K}$ and $f$, the existence and multiplicity of nontrivial solutions for the above problem are obtained. Our results cover the degenerate case in the $p(\cdot )$ fractional setting.