Year: 2024
Author: Juan Zhao, Pengxiu Yu
Journal of Partial Differential Equations, Vol. 37 (2024), Iss. 4 : pp. 402–416
Abstract
Let Ω be a smooth bounded domian in \mathbb{R}^2 , H^1_0 (Ω) be the standard Sobolev space, and λ_f (Ω) be the first weighted eigenvalue of the Laplacian, namely, \lambda_f(\Omega)=\inf\limits_{u\in H^1_0(\Omega),\int_{\Omega}u^2{\rm dx}=1}\int_{\Omega}|\nabla u|^2f{\rm dx},where f is a smooth positive function on Ω. In this paper, using blow-up analysis, we prove\sup\limits_{u\in H^1_0(\Omega),\int_{\Omega}|\nabla u|^2f{\rm dx}\le 1}\int_{\Omega}e^{4\pi fu^2(1+\alpha||u||^2_2)}{\rm dx}<+\inftyfor any 0≤α<λ_f (Ω). Furthermore, extremal functions for the above inequality exist when α>0 is chosen sufficiently small.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/jpde.v37.n4.3
Journal of Partial Differential Equations, Vol. 37 (2024), Iss. 4 : pp. 402–416
Published online: 2024-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 15
Keywords: Trudinger-Moser inequality extremal functions blow-up analysis.