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Title:  Multiple boundary peak solutions for some singularly perturbed Neumann problems 
Authors:  Winter, M Gui, C Wei, J 
Keywords:  Multiple boundary spikes Nonlinear elliptic equations 
Publication Date:  2000 
Publisher:  Elsevier 
Citation:  Ann Inst Henri Poincare Anal Nonl. 17(2000): 4782 
Abstract:  We consider the problem \left \{
\begin{array}{rcl} \varepsilon^2 \Delta u  u + f(u) = 0 & \mbox{ in }& \ \Omega\\ u > 0 \ \mbox{ in} \ \Omega, \ \frac{\partial u}{\partial \nu} = 0 & \mbox{ on }& \ \partial\Omega,
\end{array} \right. where \Omega is a bounded smooth domain in R^N, \varepsilon>$ is a small parameter and f is a superlinear, subcritical nonlinearity. It is known that this equation possesses boundary spike solutions such that
the spike concentrates, as \varepsilon approaches zero, at a critical point of the mean curvature function H(P), P \in \partial \Omega . It is also known that this equation has multiple boundary spike solutions at multiple nondegenerate critical points of H(P) or multiple local maximum points of H(P). In this paper, we prove that for any fixed positive integer $K$ there exist boundary $Kpeak$ solutions at a local minimum point of $H(P)$. This implies that for any smooth and bounded domain there always exist boundary $Kpeak$ solutions.
We first use the LiapunovSchmidt method to reduce the problem to finite dimensions.
Then we use a maximizing procedure to obtain multiple boundary spikes. 
URI:  http://bura.brunel.ac.uk/handle/2438/560 
DOI:  http://dx.doi.org/10.1016/S02941449(99)001043 
Appears in Collections:  School of Information Systems, Computing and Mathematics Research Papers Mathematical Science

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