On some elliptic equation in the whole euclidean space $ \mathbb{R}^2 $ with nonlinearities having new exponential growth condition

Abstract

In this paper, we study some elliptic equation defined in \begin{document}$ \mathbb{R}^2 $\end{document} and involving a nonlinearity with new exponential growth condition including the doubly exponential growth at infinity. For that aim, we start by extending some new Trudinger-Moser type inequalities defined on the unit ball of different classes of weighted Sobolev spaces established by B. Ruf and M. Calanchi to the whole space \begin{document}$ \mathbb{R}^2. $\end{document}