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

TL;DR

This paper investigates the existence, non-existence, and asymptotic behavior of solutions to $N$-Laplacian equations with critical exponential growth in the entire space, using a non-variational approach.

Contribution

It introduces a non-variational method to analyze solutions for elliptic problems with critical Trudinger-Moser growth, extending the literature to broader cases.

Findings

Established bounds for solutions in $L^ty$ norm despite failure of Sobolev embedding.

Proved existence and non-existence results for solutions under various conditions.

Explored asymptotic properties of solutions in the whole space.

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 $\|u_{\lambda}\|_{L^\infty(\mathbb{R}^N)} \leq C \|u\|_{W^{1,N}(\mathbb{R}^N)}^{\Theta}$ (for some $\Theta>0$), when $u$ is a solution. To conclude, we explore some asymptotic properties.