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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
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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 infinitely many solutions, prescribed boundary mean curvature, variational reduction
MSC Classifications: 35J25, 35J65, 35J67 show english descriptions Boundary value problems for second-order elliptic equations
Nonlinear boundary value problems for linear elliptic equations
Boundary values of solutions to elliptic equations
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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