The behavior of solutions of a parametric weighted $(p,q)$-Laplacian equation

We study the behavior of solutions for the parametric equation

$$-{\mathrm{\Delta }}_p^{a_1}u(z)-{\mathrm{\Delta }}_q^{a_2}u(z)=\lambda |u(z){|}^{q-2}u(z)+f(z,u(z))\text{in}\mathrm{\Omega },\lambda >0,$$

under Dirichlet condition, where $\mathrm{\Omega }\subseteq {\mathbb{R}}^N$ is a bounded domain with a $C^2$-boundary $\mathrm{\partial }\mathrm{\Omega }$, $a_1,a_2\in L^{\mathrm{\infty }}(\mathrm{\Omega })$ with $a_1(z),a_2(z)>0$ for a.a. $z\in \mathrm{\Omega }$, $p,q\in (1,\mathrm{\infty })$ and ${\mathrm{\Delta }}_p^{a_1},{\mathrm{\Delta }}_q^{a_2}$ are weighted versions of $p$-Laplacian and $q$-Laplacian. We prove existence and nonexistence of nontrivial solutions, when $f(z,x)$ asymptotically as $x\to \pm \mathrm{\infty }$ 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.