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On the $ Q$-curvature problem on $ \mathbb{S}^3$


Authors: Ruilun Cai and Sanjiban Santra
Journal: Proc. Amer. Math. Soc. 145 (2017), 119-133
MSC (2010): Primary 35G20, 35A01, 53C21
DOI: https://doi.org/10.1090/proc/13271
Published electronically: August 29, 2016
MathSciNet review: 3565365
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Abstract: Let $ P_{\mathbb{S}^3}=\Delta _0^2+ \frac {1}{2}\Delta _0- \frac {15}{16}$ denote the Panietz operator on the standard sphere $ \mathbb{S}^3$. In this paper, we study the following fourth order elliptic equation with a nonlinear term of negative power type:

$\displaystyle P_{\mathbb{S}^3} u = -\frac {1}{2}Qu^{-7}$$\displaystyle \mbox { on } \mathbb{S}^3. $

Here $ Q$ is a prescribed smooth function on $ \mathbb{S}^3$ which is assumed to be a smooth bounded positive function. We prove the existence of positive solutions to the equation under a non-degeneracy assumption on $ Q$.

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Additional Information

Ruilun Cai
Affiliation: DBS Bank, Marina Bay Financial Centre Tower 3, 12 Marina Boulevard, 018982 Singapore
Email: ruiluncai@dbs.com

Sanjiban Santra
Affiliation: Department of Basic Mathematics, Centro de Investigacióne en Mathematicás, Guanajuato, México
Email: sanjiban@cimat.mx

DOI: https://doi.org/10.1090/proc/13271
Keywords: Fourth order, negative exponent, existence, degree theory
Received by editor(s): February 27, 2016
Published electronically: August 29, 2016
Additional Notes: The second author acknowledges funding from LMAP UMR CNRS 5142, Université Pau et des Pays de l’Adour.
Communicated by: Yingfei Yi
Article copyright: © Copyright 2016 American Mathematical Society