  We investigate the existence and multiplicity of solutions to the following
$p(x)$-Laplacian problem in $\mathbb{R}^{N}$ via critical point theory
\begin{equation*} \left\{ \begin{array}{l} -\bigtriangleup
_{p(x)}u+V(x)\left\vert u\right\vert ^{p(x)-2}u=f(x,u),\text{ in }
\mathbb{R}^{N}, \\ u\in W^{1,p(\cdot )}(\mathbb{R}^{N}). \end{array} \right.
\end{equation*} We propose a new set of growth conditions which matches the
variable exponent nature of the problem. Under this new set of assumptions, we
manage to verify the Cerami compactness condition. Therefore, we succeed in
proving the existence of multiple solutions to the above problem without the
well-known Ambrosetti--Rabinowitz type growth condition. Meanwhile, we could
also characterize the pointwise asymptotic behaviors of these solutions. In our
main argument, the idea of localization, decomposition of the domain,
regularity of weak solutions and comparison principle are crucial ingredients
among others.
