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:

$$\begin{aligned} \left\{ \begin{array}{ll} {\mathcal {K}}\left( \int _{{\mathbb {R}}^{2N}}\frac{1}{p(x,y)}\frac{|\varphi (x)-\varphi (y)|^{p(x,y)}}{|x-y|^{N+s(x,y){p(x,y)}}} \,dx\,dy\right) (-\Delta )^{s(\cdot )}_{p(\cdot )}\varphi (x) =f(x,\varphi ) \quad \text{ in } \Omega , \\ \varphi =0 \quad \text{ on } {\mathbb {R}}^N\backslash \Omega . \end{array} \right. \end{aligned}$$

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.