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.

In this article, we investigate the Kirchhoff-Schrödinger-Poisson type systems on the Heisenberg group of the following form:

$$\{\begin{array}{ll}-(a+b{\int }_{\mathrm{\Omega }}|{\mathrm{\nabla }}_{H}u{|}^{p}d\xi ){\mathrm{\Delta }}_{H,p}u-\mu \varphi |u{|}^{p-2}u=\lambda |u{|}^{q-2}u+|u{|}^{Q^{\ast }-2}u& \text{in}\mathrm{\Omega },\\ -{\mathrm{\Delta }}_H\varphi =|u{|}^p& \text{in}\mathrm{\Omega },\\ u=\varphi =0& \text{on}\mathrm{\partial }\mathrm{\Omega },\end{array}$$

where $a,b$ are positive real numbers, $\mathrm{\Omega }\subset {\mathbb{H}}^N$ is a bounded region with smooth boundary, $1<p<Q$, $Q=2N+2$ is the homogeneous dimension of the Heisenberg group ${\mathbb{H}}^N$, $Q^{\ast }=\frac{pQ}{Q-p}$, $q\in (2p,Q^{\ast })$ and ${\mathrm{\Delta }}_{H,p}u=\text{div}(|{\mathrm{\nabla }}_{H}u{|}^{p-2}{\mathrm{\nabla }}_{H}u)$ is the $p$-horizontal Laplacian. Under some appropriate conditions for the parameters $\mu$ and $\lambda$, we establish existence and multiplicity results for the system above. To some extent, we generalize the results of An and Liu (Israel J. Math., 2020) and Liu et al. (Adv. Nonlinear Anal., 2022).