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


Authors: Djairo G. de Figueiredo, João Marcos B. do Ó and Ederson Moreira dos Santos
Journal: Proc. Amer. Math. Soc. 144 (2016), 3369-3380
MSC (2010): Primary 35J15; Secondary 46E35
DOI: https://doi.org/10.1090/proc/13114
Published electronically: March 30, 2016
MathSciNet review: 3503705
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Abstract: In this paper we study some weighted Trudinger-Moser type problems, namely

$\displaystyle \displaystyle {s_{F,h} = \sup _{u \in H, \, \Vert u\Vert _H =1 } \int _{B} F(u) h(\vert x\vert) dx},$    

where $ B \subset {\mathbb{R}}^2$ represents the open unit ball centered at zero in $ {\mathbb{R}}^2$ and $ H$ stands either for $ H^1_{0, \textrm {rad}}(B)$ or $ H^1_{\textrm {rad}}(B)$. We present the precise balance between $ h(r)$ and $ F(t)$ that guarantees $ s_{F,h}$ to be finite. We prove that $ s_{F,h}$ is attained up to the $ h(r)$-radially critical case. In particular, we solve two open problems in the critical growth case.

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Additional Information

Djairo G. de Figueiredo
Affiliation: IMECC-UNICAMP, Caixa Postal 6065, CEP 13083-859, Campinas, SP, Brazil
Email: djairo@ime.unicamp.br

João Marcos B. do Ó
Affiliation: Departamento de Matemática, UFPB, CEP 58051-900, João Pessoa, PB, Brazil
Email: jmbo@pq.cnpq.br

Ederson Moreira dos Santos
Affiliation: ICMC-USP. Caixa Postal 668, CEP 13560-970, São Carlos, SP, Brazil
Email: ederson@icmc.usp.br

DOI: https://doi.org/10.1090/proc/13114
Keywords: Trudinger-Moser inequality, H{\'e}non type equation, critical growth problem
Received by editor(s): August 14, 2014
Published electronically: March 30, 2016
Additional Notes: The first author’s research was partially supported by CAPES and CNPq
The second author’s research was partially supported by CAPES, CNPq and INCT-MAT
The third author’s research was partially supported by CAPES, CNPq, and FAPESP
Communicated by: Alexander Iosevich
Article copyright: © Copyright 2016 American Mathematical Society