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

Abstract

In this paper, we are interested in studying the existence or non-existence of solutions for a class of elliptic problems involving the $N$-Laplacian operator in the whole space. The nonlinearity considered involves critical Trudinger-Moser growth. Our approach is non-variational, and in this way, we can 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, we establish $\Vert u_{\lambda }{\Vert }_{L^{\infty }({\mathbb{R}}^N)}\le C\Vert u{\Vert }_{W^{1,N}({\mathbb{R}}^N)}^{\Theta }$ (for some $\Theta >0$), when $u$ is a solution. To conclude, we explore some asymptotic properties.

37 pages