  This paper deals with the existence and multiplicity of solutions for the
generalized $(p, q)$-Laplacian equation \begin{align*} &-{\text{ div}}(A(x,
u)|\nabla u|^{p-2}\nabla u) +\frac1p A_t(x, u)|\nabla u|^p -{\text{ div}}(B(x,
u)|\nabla u|^{q-2}\nabla u) \\ &\quad\qquad+\frac1q B_t(x, u)|\nabla u|^q +
V(x)|u|^{p-2} u+ W(x)|u|^{q-2} u= g(x, u)\quad\qquad\mbox{ in } \mathbb{R}^N,
\end{align*} where $1<q\le p< N$, $A,
B:\mathbb{R}^N\times\mathbb{R}\to\mathbb{R}$ are suitable $C^1$ Carath\'eodory
functions with $A_t(x, u)=\frac{\partial A}{\partial t}(x, u), B_t(x,
u)=\frac{\partial B}{\partial t}(x, u)$, $V, W:\mathbb{R}^N\to\mathbb{R}$ are
proper ``weight functions" and $g:\mathbb{R}^N\times\mathbb{R}\to\mathbb{R}$ is
a Carath\'eodory map. Notwithstanding the occurrence of some coefficients which
rely upon the solution itself makes the use of variational techniques more
challenging, under suitable assumptions on the involved functions, we are able
to exploit the variational nature of our problem. In particular, the existence
of a nontrivial solution is derived via a generalized version of the
Ambrosetti-Rabinowitz Mountain Pass Theorem, based on a weaker version of the
classical Cerami-Palais-Smale condition. Finally, the multiplicity result,
which is thoroughly new also even in the simpler case $q=p$, is gained under
symmetry assumptions and a sharp decomposition of the ambient space.
