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In this paper, we are interested in studying the existence or non-existenceof solutions for a class of elliptic problems involving the $N$-Laplacianoperator in the whole space. The nonlinearity considered involves criticalTrudinger-Moser growth. Our approach is non-variational, and in this way, wecan address a wide range of problems not yet contained in the literature. Even$W^{1,N}(\mathbb{R}^N)\hookrightarrow L^\infty(\mathbb{R}^N)$ failing, weestablish $\|u_{\lambda}\|_{L^\infty(\mathbb{R}^N)} \leq C\|u\|_{W^{1,N}(\mathbb{R}^N)}^{\Theta}$ (for some $\Theta>0$), when $u$ is asolution. To conclude, we explore some asymptotic properties.

Existence, nonexistence, and asymptotic behavior of solutions for  $N$-Laplacian equations involving critical exponential growth in the whole  $\mathbb{R}^N$