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On Multiple Mixed Interior and Boundary Peak Solutions for Some Singularly Perturbed Neumann Problems

 Printed: Jun 2000
  • Changfeng Gui
  • Juncheng Wei
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We consider the problem \begin{equation*} \begin{cases} \varepsilon^2 \Delta u - u + f(u) = 0, u > 0 & \mbox{in } \Omega\\ \frac{\partial u}{\partial \nu} = 0 & \mbox{on } \partial\Omega, \end{cases} \end{equation*} where $\Omega$ is a bounded smooth domain in $R^N$, $\ve>0$ is a small parameter and $f$ is a superlinear, subcritical nonlinearity. It is known that this equation possesses multiple boundary spike solutions that concentrate, as $\epsilon$ approaches zero, at multiple critical points of the mean curvature function $H(P)$, $P \in \partial \Omega$. It is also proved that this equation has multiple interior spike solutions which concentrate, as $\ep\to 0$, at {\it sphere packing\/} points in $\Om$. In this paper, we prove the existence of solutions with multiple spikes {\it both\/} on the boundary and in the interior. The main difficulty lies in the fact that the boundary spikes and the interior spikes usually have different scales of error estimation. We have to choose a special set of boundary spikes to match the scale of the interior spikes in a variational approach.
Keywords: mixed multiple spikes, nonlinear elliptic equations mixed multiple spikes, nonlinear elliptic equations
MSC Classifications: 35B40, 35B45, 35J40 show english descriptions Asymptotic behavior of solutions
A priori estimates
Boundary value problems for higher-order elliptic equations
35B40 - Asymptotic behavior of solutions
35B45 - A priori estimates
35J40 - Boundary value problems for higher-order elliptic equations

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