  This paper deals with the existence of multiple solutions for the quasilinear
equation $-\mathrm{div}\,\mathbf{A}(x,\nabla u)| u| ^{\alpha (x)-2}u=f(x,u)$ in
$ \mathbb{R} ^{N}$, which involves a general variable exponent elliptic
operator $\mathbf{ A}$ in divergence form. The problem corresponds to double
phase anisotropic phenomena, in the sense that the differential operator has
behaviors like $ | \xi | ^{q(x)-2}\xi $ for small $| \xi | $ and like $| \xi |
^{p(x)-2}\xi $ for large $ | \xi | $, where $1<\alpha (\cdot )\leq p(\cdot
)<q(\cdot )<N$. Our aim is to approach variationally the problem by using the
tools of critical points theory in generalized Orlicz-Sobolev spaces with
variable exponent. Our results extend the previous works Azzollini, d'Avenia,
and Pomponio (2014) and Chorfi and R\u{a}dulescu (2016), from the case when
exponents $p$ and $q$ are constant, to the case when $p(\cdot )$ and $% q(\cdot
)$ are functions. We also substantially weaken some of their hypotheses
overcome the lack of compactness by using the weighting method.
