Trudinger-Moser inequalities involving fast growth and weights with strong vanishing at zero
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- by Djairo G. de Figueiredo, João Marcos B. do Ó and Ederson Moreira dos Santos PDF
- Proc. Amer. Math. Soc. 144 (2016), 3369-3380 Request permission
Abstract:
In this paper we study some weighted Trudinger-Moser type problems, namely \begin{equation*} \displaystyle {s_{F,h} = \sup _{u \in H, \| u\|_H =1 } \int _{B} F(u) h(|x|) dx}, \end{equation*} 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.References
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Additional Information
- Djairo G. de Figueiredo
- Affiliation: IMECC-UNICAMP, Caixa Postal 6065, CEP 13083-859, Campinas, SP, Brazil
- MR Author ID: 66760
- ORCID: 0000-0002-9902-244X
- Email: djairo@ime.unicamp.br
- João Marcos B. do Ó
- Affiliation: Departamento de Matemática, UFPB, CEP 58051-900, João Pessoa, PB, Brazil
- MR Author ID: 365349
- Email: jmbo@pq.cnpq.br
- Ederson Moreira dos Santos
- Affiliation: ICMC-USP. Caixa Postal 668, CEP 13560-970, São Carlos, SP, Brazil
- MR Author ID: 848409
- Email: ederson@icmc.usp.br
- 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
- © Copyright 2016 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 144 (2016), 3369-3380
- MSC (2010): Primary 35J15; Secondary 46E35
- DOI: https://doi.org/10.1090/proc/13114
- MathSciNet review: 3503705