*Mathematical Methods in the Applied Sciences*Multiple solutions for a class of non-cooperative critical nonlocal
equation system with variable exponents

In this paper, we consider a class of non-cooperative critical nonlocal
equation system with variable exponents of the form: $$
\left\{
\begin{array}{lll}
-(-\Delta)_{p(\cdot,\cdot)}^su
- |u|^{p(x)-2}u = F_u(x,u,v) +
|u|^{q(x)-2}u, \quad
&\mbox{in}\,\,\mathbb{R}^N,\\
(-\Delta)_{p(\cdot,\cdot)}^sv
+ |v|^{p(x)-2}v = F_v(x,u,v) +
|v|^{q(x)-2}u, \quad
&\mbox{in}\,\,\mathbb{R}^N,\\
u, v \in
W^{s,p(\cdot,\cdot)}(\mathbb{R}^N),
\end{array}\right. $$ where
$\nabla F = (F_u, F_v)$ is the gradient of a
$C^1$-function $F:
\mathbb{R}^N\times
\mathbb{R}^2 \rightarrow
\mathbb{R}^+$ with respect to the variable $(u, v)
\in \mathbb{R}^2$. We also assume
that$\{x \in
\mathbb{R}^N: q(x) =
p_s^\ast(x)\} \neq
\emptyset$, here
$p_s^\ast(x)=Np(x,x)/(N-sp(x,x))$ is the critical
Sobolev exponent for variable exponents. With the help of the Limit
index theory and the concentration-compactness principles for fractional
Sobolev spaces with variable exponents, we establish the existence of
infinitely many solutions for the problem under the suitable conditions
on the nonlinearity.

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