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 {equation*} {cases} -Δ_p u -Δ_N u = |u|q-2u ln|u|^2 + λ f(u) & in Ω,\\ u=0 & on ∂ Ω, {cases} {equation*} where Ω ⊂ RN is a bounded domain, N ≥ 2, $1< p< N< q$, λ > 0 is a positive real parameter. By applying variational methods, we obtain the existence of solutions.