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The present work is concerned with the following version of ChoquardLogarithmic 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) \textrm{ in } \mathbb{R}^N $ , where $a, b, \lambda >0 $, $ \max\{\frac{N}{2}, 2 \} < p< N $, $f, g: \mathbb{R}\rightarrow \mathbb{R} $ are continuous functions that behave like $\exp(\alpha |u|^{\frac{N}{N-1}}) $ at infinity, for $ \alpha >0 $, and that haspolynomial growth, respectively, and $ G(s)=\int\limits_{0}^{s}g(\tau)d\tau $.We prove the existence of a nontrivial solution at the mountain pass level anda nontrivial ground state solution. Also, using a version of the SymmetricMountain-Pass Theorem, we get infinitely many solutions.

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