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

In this paper, we are interested in studying the existence or non-existence of solutions for a class elliptic problems involving N-Laplacian operator whole space. The nonlinearity considered involves critical Trudinger–Moser growth. Our approach is non-variational, and way can address wide range not yet contained literature. Even $$W^{1,N}({\mathbb {R}} ^N)\hookrightarrow L^\infty ({\mathbb ^N)$$ failing, establish $$\Vert u\Vert _{L^\infty {R}}^N)} \le C \Vert _{W^{1,N}({\mathbb {R}}^N)}^{\Theta }$$ (for some $$\Theta >0$$ ), when u solution. To conclude, explore asymptotic properties.