Existence and nonexistence of maximizers for variational problems associated with Trudinger–Moser type inequalities in $${\mathbb{R}^N}$$

Abstract

We discuss the existence of a maximizer for a maximizing problem associated with the Trudinger–Moser type inequality in \({\mathbb{R}^N(N\geq 2)}\). Different from the bounded domain case, we obtain both of the existence and the nonexistence results. The proof requires a careful estimate of the maximizing level with the aid of normalized vanishing sequences.