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 R N (N ≥ 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. M. Ishiwata (B)

On the effect of equivalent constraints on a maximizing problem associated with the Sobolev type embeddings in $${\mathbb {R}}^N$$ R N

“…in [3,4,8]. As for the attainability of the supremum d N ,β in unbounded domains, the first author proved the following fact [5]. Also for the case (a), the attainability of d N ,β N is proved by Li-Ruf [7].…”

“…Just after the publication of the paper [5], the second author pointed out that, even in the higher dimensional case, the attainability of d N ,β heavily depends on the value β if one replaces the normalizing condition ∇u N N + u N N = 1 by ∇u N + u N = 1. This result suggests that the attainability of the supremum value depends delicately on the choice of normalizing conditions even if conditions are equivalent.…”

Extremal Functions for Trudinger-Moser Type Inequalities in ℝ^N

“…In the case α = α N and N ≥ 3, the existence of extremal functions for the supremum in (1.2) was obtained by Li-Ruf [11]; while in the case 0<α<α N , the existence result was proved by Ishiwata [13]. From now on, we assume N ≥ 2.…”

Vanishing-concentration-compactness alternative for the Trudinger–Moser inequality in ℝN

“…Our aim is to clarify the influence of the constraint S a,b := {u ∈ W 1,N (R N ) | ∇u a N + u b N = 1} on concentration phenomena of (spherically symmetric and non-increasing) maximizing sequences for the Trudinger-Moser supremum In the 2-dimensional case, the study of the attainability of the supremum d 2,α is due to Ruf [19] and Ishiwata [13]. Roughly speaking, from the delicate analysis carried out in [19,13], we can deduce that, given a (spherically symmetric and nonincreasing) maximizing sequence {u j } j ⊂ W 1,2 (R 2 ) for d 2,α with 0 < α ≤ α N , the following alternative occurs: either the weak limit u in W 1,2 (R 2 ) of the maximizing sequence {u j } j is non-trivial (compactness) and it is a maximizer for d 2,α or u = 0. In the latter case, the loss of compactness can be caused by The proper understanding of the above alternative was a priori not obvious.…”

“…In the latter case, the loss of compactness can be caused by The proper understanding of the above alternative was a priori not obvious. However, the most valuable results obtained in [19,13] cannot be summarized in this way and are clearly more involved. In the critical case α = α 2 = 4π, as showed in [13], it is possible to rule out vanishing behaviors of maximizing sequences for d 2,4π and the most hard and inspiring part of the result in [19] is to exclude concentration phenomena.…”