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

Abstract

This paper deals with a class of nonlocal variable $s(.)$-order fractional $p(.)$-Kirchhoff type equations: \begin{eqnarray*} \left\{ \begin{array}{ll} \mathcal{K}\left(\int_{\mathbb{R}^{2N}}\frac{1}{p(x,y)}\frac{|φ(x)-φ(y)|^{p(x,y)}}{|x-y|^{N+s(x,y){p(x,y)}}} \,dx\,dy\right)(-Δ)^{s(\cdot)}_{p(\cdot)}φ(x) =f(x,φ) \quad \mbox{in }Ω, \\ φ=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.