Abstract
Let varOmega subset mathbb{R}^{2} be a smooth bounded domain, W_{0}^{1,2}(varOmega ) be the standard Sobolev space. Assuming certain conditions on a function g:mathbb{R}rightarrow mathbb{R}, we derive a modified singular Trudinger–Moser inequality, which was originally established by Adimurthi and Sandeep (Nonlinear Differ. Equ. Appl. 13:585–603, 2007), namely, \t\t\t1supu∈W01,2(Ω),∥∇u∥2≤1∫Ω(1+g(u))e4π(1−γ)u2|x|2γdx,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\sup_{u\\in W_{0}^{1,2}(\\varOmega ), \\Vert \\nabla u \\Vert _{2}\\leq 1} \\int _{\\varOmega }\\bigl(1+g(u)\\bigr)\\frac{e ^{4\\pi (1-\\gamma ) u^{2}}}{ \\vert x \\vert ^{2\\gamma }}\\,dx, $$\\end{document} where 0<gamma <1. Following Yang and Zhu (J. Funct. Anal. 272:3347–3374, 2017), we prove that the extremal functions for the supremum in (1) exist. The proof is based on a blow-up analysis.
Highlights
Anal. 272:3347–3374, 2017), we prove that the extremal functions for the supremum in (1) exist
A function u0 is called an extremal function for the Trudinger–Moser inequality (2) if u0 belongs to W01,2(Ω), ∇u0 2 ≤ 1 and eαu20 dx =
In order to prove the critical singular Trudinger–Moser inequality, we firstly discuss the existence of extremal functions for a subcritical one, which is based on a direct method variation
Summary
Using a rearrangement argument and a change of variables, Adimurthi–Sandeep [2] generalized the Trudinger–Moser inequality (1) to a singular version as follows: e4π(1–γ )u2 sup dx < ∞. This inequality is sharp in the sense that all integrals are still finite when α > 1 – γ , but the supremum is infinity. We are aim to prove two main results: One is to explain the new supremum is finite; the other is to discuss the existence of extremals for such functionals. In order to prove the critical singular Trudinger–Moser inequality, we firstly discuss the existence of extremal functions for a subcritical one, which is based on a direct method variation. We refer to Adimurthi and Druet [1], Carleson–Chang [5], Li [15], Struwe [24], Adimurthi and Struwe [3], Iula and Mancini [13], Yang [28], Lu and Yang [18], respectively
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
