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# Infinitely Many Solutions for the Prescribed Boundary Mean Curvature Problem in $\mathbb B^N$

Published:2012-12-29
Printed: Aug 2013
• Liping Wang,
Department of Mathematics, East China Normal University, Shanghai, 200241, China
• Chunyi Zhao,
Department of Mathematics, East China Normal University, Shanghai, 200241, China
 Format: LaTeX MathJax PDF

## Abstract

We consider the following prescribed boundary mean curvature problem in $\mathbb B^N$ with the Euclidean metric: $\begin{cases} \displaystyle -\Delta u =0,\quad u\gt 0 &\text{in }\mathbb B^N, \\[2ex] \displaystyle \frac{\partial u}{\partial\nu} + \frac{N-2}{2} u =\frac{N-2}{2} \widetilde K(x) u^{2^\#-1} \quad & \text{on }\mathbb S^{N-1}, \end{cases}$ where $\widetilde K(x)$ is positive and rotationally symmetric on $\mathbb S^{N-1}, 2^\#=\frac{2(N-1)}{N-2}$. We show that if $\widetilde K(x)$ has a local maximum point, then the above problem has infinitely many positive solutions that are not rotationally symmetric on $\mathbb S^{N-1}$.
 Keywords: infinitely many solutions, prescribed boundary mean curvature, variational reduction
 MSC Classifications: 35J25 - Boundary value problems for second-order elliptic equations 35J65 - Nonlinear boundary value problems for linear elliptic equations 35J67 - Boundary values of solutions to elliptic equations

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