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Uniqueness of finite total curvatures and the structure of radial solutions for nonlinear elliptic equations


Authors: Jann-Long Chern, Zhi-You Chen and Yong-Li Tang
Journal: Trans. Amer. Math. Soc. 363 (2011), 3211-3231
MSC (2010): Primary 35J60; Secondary 34A12
DOI: https://doi.org/10.1090/S0002-9947-2011-05192-5
Published electronically: January 25, 2011
MathSciNet review: 2775804
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Abstract: In this article, we are concerned with the semilinear elliptic equation

$\displaystyle \Delta u+K(\vert x\vert)\vert u\vert^{p-1}u=0\quad\textrm{in} \mathbf{R}^n\setminus\{\mathbf{0}\},$

where $ n>2$, $ p>1$, and $ K(\vert x\vert)>0$ in $ \mathbf{R}^n$. The correspondence between the initial values of regularly positive radial solutions of the above equation and the associated finite total curvatures will be derived. In addition, we also conduct the zeros of radial solutions in terms of the initial data under specific conditions on $ K$ and $ p$. Furthermore, based on the Pohozaev identity and openness for the regions of initial data corresponding to certain types of solutions, we obtain the whole structure of radial solutions depending on various situations.


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Additional Information

Jann-Long Chern
Affiliation: Department of Mathematics, National Central University, Chung-Li 32001, Taiwan
Email: chern@math.ncu.edu.tw

Zhi-You Chen
Affiliation: Department of Mathematics, National Central University, Chung-Li 32001, Taiwan
Address at time of publication: Department of Mathematics, National Tsing Hua University, Hsin-Chu 30013, Taiwan
Email: zhiyou@math.ncu.edu.tw

Yong-Li Tang
Affiliation: Department of Mathematics, National Central University, Chung-Li 32001, Taiwan
Email: tangyl@math.ncu.edu.tw

DOI: https://doi.org/10.1090/S0002-9947-2011-05192-5
Received by editor(s): March 19, 2008
Received by editor(s) in revised form: August 7, 2009
Published electronically: January 25, 2011
Additional Notes: The work of the first author was partially supported by the National Science Council of Taiwan
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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