| Summary: | This work deals with the nonlocal $p(x)$-Laplacian equations in $R^{N}$ with non-variational form
\begin{align*}
\left\{\begin{aligned}
&A(u)\big(-\Delta_{p(x)}u+|u|^{p(x)-2}u\big)=B(u)f(x,u) \text{in}R^{N},\\
&u\in W^{1, p(x)}(R^{N}),
\end{aligned}
\right.\end{align*}
and with the variational form
\begin{align*}
\left\{\begin{aligned}
& a\Big(\int_{R^{N}}\frac{\vert \nabla u\vert^{p(x)}+\vert u\vert^{p(x)}}{p(x)}dx\Big)(-\Delta_{p(x)}u+|u|^{p(x)-2}u)&\\
&=B\Big(\int_{R^{N}}F(x, u)dx \Big)f(x, u) \text{in} R^{N},&\\
&u\in W^{1, p(x)}(R^{N}),
\end{aligned}\right.
\end{align*}
where $F(x,t)=\int_{0}^{t}f(x,s)ds$, and $a$ is allowed to be singular at zero. Using $(S_{+})$ mapping theory and the variational method, some results on existence and multiplicity for the problems in $R^{N}$ are obtained.
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