  We study the behavior of solutions for the parametric equation
$$-\Delta_{p}^{a_1} u(z)-\Delta_{q}^{a_2} u(z)=\lambda |u(z)|^{q-2}
u(z)+f(z,u(z)) \quad \mbox{in } \Omega,\, \lambda >0,$$ under Dirichlet
condition, where $\Omega \subseteq \mathbb{R}^N$ is a bounded domain with a
$C^2$-boundary $\partial \Omega$, $a_1,a_2 \in L^\infty(\Omega)$ with
$a_1(z),a_2(z)>0$ for a.a. $z \in \Omega$, $p,q \in (1,\infty)$ and
$\Delta_{p}^{a_1},\Delta_{q}^{a_2}$ are weighted versions of $p$-Laplacian and
$q$-Laplacian. We prove existence and nonexistence of nontrivial solutions,
when $f(z,x)$ asymptotically as $x \to \pm \infty$ can be resonant. In the
studied cases, we adopt a variational approach and use truncation and
comparison techniques. When $\lambda$ is large, we establish the existence of
at least three nontrivial smooth solutions with sign information and ordered.
Moreover, the critical parameter value is determined in terms of the spectrum
of one of the differential operators.
