*Mathematical Methods in the Applied Sciences*A variable-order fractional $p(\cdot)$-Kirchhoff type
problem in $\mathbb{R}^{N}$

This paper is concerned with the existence and multiplicity of solutions
for the following variable $s(\cdot)$-order fractional
$p(\cdot)$-Kirchhoff type problem
\begin{equation*}
\left\{\begin{array}{ll}
M\left(\displaystyle\iint_{\mathbb
R^{2N}}\frac{1}{p(x,y)}\displaystyle{\frac{|v(x)-v(y)|^{p(x,y)}}{|x-y|^{N+p(x,y)s(x,y)}}}dxdy\right)(-\Delta)^{s(\cdot)}_{p(\cdot)}v(x)+|v(x)|^{\overline{p}(x)-2}v(x)
=\mu g(x,v)\ \
{\rm
in}~\mathbb{R}^{N},\\
v\in
W^{s(\cdot),p(\cdot)}(\mathbb{R}^{N}),
\end{array}\right.
\end{equation*} where $N>p(x,y)s(x,y)$
for any
$(x,y)\in\mathbb{R}^{N}\times\mathbb{R}^{N}$,
$(-\Delta)^{s(\cdot)}_{p(\cdot)}$
is a variable $s(\cdot)$-order
$p(\cdot)$-fractional Laplace operator with
$s(\cdot):\mathbb
R^{2N}\to(0,1)$ and
$p(\cdot):\mathbb
R^{2N}\to(1,\infty)$,
$\overline{p}(x)=p(x,x)$ for
$x\in\mathbb{R}^{N}$, and $M$
is a continuous Kirchhoff-type function, $g(x,v)$ is a
Carath\’{e}odory function,
$\mu>0$ is a parameter. We obtain that
there are at least two distinct solutions for the above problem by
applying the generalized abstract critical point theorem. Under the
weaker conditions, we also show the existence of one solution and
infinitely many solutions by using the mountain pass lemma and fountain
theorem, respectively. In particular, the new compact embedding result
of the space $
W^{s(\cdot),p(\cdot)}(\mathbb{R}^{N})$
into
$L^{q(\cdot)}_{a(x)}(\mathbb{R}^{N})$
will be used to overcome the lack of compactness in
$\mathbb{R}^N$. The main feature and difficulty of
this paper is the presence of a double non-local term involving two
variable parameters.

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