Optimizers for the singular Trudinger–Moser inequalities in the critical case in

Abstract

The main purpose of this paper is to study the existence of extremal functions for the singular Trudinger–Moser inequalities in the critical case in R2. More precisely, let 0≤β<2 and denote 0true81.60004pt74.0ptTMa,b,β()α=sup∇u2a+u2b≤1∫double-struckR2[]expα()1−β2false|ufalse|2−1dx||xβ,then we will prove in this article that there exists ε>0 such that TMa,2,β4π can be achieved for all 0≤β<ε, 2−ε<a<2+ε.

On an effect of inhomogeneous constraints for a maximizing problem of the Sobolev embedding associated with the space of bounded variation

“…Attainability problems associated with those inequalities also have been investigated in rich literature. Among others, we refer to [4,5,12,17,18] and related works [8,9,10,11,13,15], in which similar problems to D α were studied. This paper is organized as follows.…”

On the effect of inhomogeneous constraints for a maximizing problem associated with the Sobolev embedding of the space of functions of bounded variation

“…Attainability problems associated with those inequalities also have a rich literature. Among others, we refer to [IIW, I, LR, OST, R] and related works [L1,L2,LLZ,Li,LY,N2], in which problems similar to D α were studied.…”