Resonance problems for the $(p-q)$-Laplacian type equations

In this paper, we consider resonance problems at an arbitrary eigenvalue of the following ($p-q$)-Laplacian equations \begin{equation*} \begin{cases} -\Delta_{p}u-\nu\Delta_{q}u =\lambda|u|^{p-2}u+g(u)-h(x), & \text{in}\ \Omega,\\ u=0, & \text{on}\ \partial\Omega. \end{cases} \end{equation*} Under some assumptions on $g$ and $h$, we establish the existence of nontrivial solutions with the help of the Saddle Point theorem.