$(p,N)-$Choquard logarithmic equation involving a nonlinearity with exponential critical growth: existence and multiplicity

The present work is concerned with the following version of Choquard Logarithmic equations $ -\Delta_p u -\Delta_N u + a|u|^{p-2}u + b|u|^{N-2}u + \lambda (\ln|\cdot|\ast G(u))g(u) = f(u) $\text{ in }$ $\mathbb{R}$^N $,where$ a, b, \lambda >0 $,$ \max{\frac{N}{2}, 2 } < p< N $,$f, g: $\mathbb{R}$ \rightarrow $\mathbb{R}$ $arecontinuousfunctionsthatbehavelike$ \exp(\alpha |u|^{\frac{N}{N-1}}) $atinfinity,for$ \alpha >0 $, and that has polynomial growth, respectively, and $G(s)={\int }_0^{s}g(\tau )d\tau$. We prove the existence of a nontrivial solution at the mountain pass level and a nontrivial ground state solution. Also, using a version of the Symmetric Mountain-Pass Theorem, we get infinitely many solutions.