Trudinger-Moser inequalities involving fast growth and weights with strong vanishing at zero

In this paper we study some weighted Trudinger-Moser type problems, namely sF,h=supu∈H,‖u‖H=1∫BF(u)h(|x|)dx,\begin{equation*} \displaystyle {s_{F,h} = \sup _{u \in H, \, \| u\|_H =1 } \int _{B} F(u) h(|x|) dx}, \end{equation*} where B⊂R2B \subset {\mathbb R}^2 represents the open unit ball centered...

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Bibliographic Details
Published in:Proceedings of the American Mathematical Society 2016-08, Vol.144 (8), p.3369-3380
Main Authors: de Figueiredo, Djairo G., do Ó, João Marcos B., dos Santos, Ederson Moreira
Format: Article
Language:English
Subjects:
B. ANALYSIS
Research article
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Summary:In this paper we study some weighted Trudinger-Moser type problems, namely sF,h=supu∈H,‖u‖H=1∫BF(u)h(|x|)dx,\begin{equation*} \displaystyle {s_{F,h} = \sup _{u \in H, \, \| u\|_H =1 } \int _{B} F(u) h(|x|) dx}, \end{equation*} where B⊂R2B \subset {\mathbb R}^2 represents the open unit ball centered at zero in R2{\mathbb R}^2 and HH stands either for H0,rad1(B)H^1_{0, \textrm {rad}}(B) or Hrad1(B)H^1_{\textrm {rad}}(B). We present the precise balance between h(r)h(r) and F(t)F(t) that guarantees sF,hs_{F,h} to be finite. We prove that sF,hs_{F,h} is attained up to the h(r)h(r)-radially critical case. In particular, we solve two open problems in the critical growth case.
ISSN:0002-9939
1088-6826
DOI:10.1090/proc/13114