  We consider a parametric nonautonomous $(p, q)$-equation with unbalanced
growth as follows
  \begin{align*}
  \left\{ \begin{aligned} &-\Delta_p^\alpha u(z)-\Delta_q u(z)=\lambda \vert
u(z)\vert^{\tau-2}u(z)+f(z, u(z)), \quad \quad \hbox{in }\Omega,\\
&u|_{\partial \Omega}=0, \end{aligned} \right. \end{align*} where $\Omega
\subseteq \mathbb{R}^N$ be a bounded domain with Lispchitz boundary
$\partial\Omega$, $\alpha \in L^{\infty}(\Omega)\backslash \{0\}$, $a(z)\geq 0$
for a.e. $z \in \Omega$, $ 1<\tau< q<p<N$ and $\lambda>0$. In the reaction
there is a parametric concave term and a perturbation $f(z, x)$. Under the
minimal conditions on $f(z, 0)$, which essentially restrict its growth near
zero, by employing variational tools, truncation and comparison techniques, as
well as critical groups, we prove that for all small values of the parameter
$\lambda>0$, the problem has at least three nontrivial bounded solutions
(positive, negative, nodal), which are ordered and asymptotically vanish as
$\lambda \rightarrow 0^{+}$.
