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Infinitely Many Solutions for the Prescribed Boundary Mean Curvature Problem in 𝔹N

Published online by Cambridge University Press:  20 November 2018

Liping Wang  and
Chunyi Zhao
Liping Wang
Affiliation:
Department of Mathematics, East China Normal University, Shanghai, 200241, China, e-mail: lpwang@math.ecnu.edu.cn, cyzhao@math.ecnu.edu.cn
Chunyi Zhao
Affiliation:
Department of Mathematics, East China Normal University, Shanghai, 200241, China, e-mail: lpwang@math.ecnu.edu.cn, cyzhao@math.ecnu.edu.cn
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Abstract

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We consider the prescribed boundary mean curvature problem in ${{\mathbb{B}}^{N}}$ with the Euclidean metric

$$\{_{\frac{\partial u}{\partial v}+\frac{N-2}{2}u=\frac{N-2}{2}\tilde{K}\left( x \right){{u}^{{{2}^{\#-1}}}}\,\,\,\,\,\,\text{on}{{\mathbb{S}}^{N-1}},}^{-\Delta u=0,\,\,\,\,\,\,u>0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{in}{{\mathbb{B}}^{N}},}$$

where $\tilde{K}\left( x \right)$ is positive and rotationally symmetric on ${{\mathbb{S}}^{N-1}},{{2}^{\#}}=\frac{2\left( N-1 \right)}{N-2}$. We show that if $\tilde{K}\left( x \right)$ has a local maximum point, then this problem has infinitely many positive solutions that are not rotationally symmetric on ${{\mathbb{S}}^{N-1}}$.

Keywords

35J2535J6535J67infinitely many solutionsprescribed boundary mean curvaturevariational reduction
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 65 , Issue 4 , 01 August 2013 , pp. 927 - 960
Copyright
Copyright © Canadian Mathematical Society 2013

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