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

Abstract

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.