Existence of solutions for the $(p,N)$-Laplacian equation with logarithmic and critical exponential nonlinearities

This paper deals with the following $(p,N)$-Laplacian equation with logarithmic and critical exponential nonlinearities. Precisely, we study the problem \begin{equation*} \begin{cases} -\Delta_p u -\Delta_N u = |u|^{q-2}u \ln|u|^2 + \lambda f(u) & \text{in }\Omega,\\ u=0 & \text{on }\partial \Omega, \end{cases} \end{equation*} where $\Omega \subset \mathbb{R}^N$ is a bounded domain, $N \geq 2$, $1< p< N< q$, $\lambda > 0$ is a positive real parameter. By applying variational methods, we obtain the existence of solutions.