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Please use this identifier to cite or link to this item: http://bura.brunel.ac.uk/handle/2438/560

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): 47-82
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 $K-peak$ solutions at a local minimum point of $H(P)$. This implies that for any smooth and bounded domain there always exist boundary $K-peak$ solutions. We first use the Liapunov-Schmidt 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/S0294-1449(99)00104-3
Appears in Collections:Mathematical Science
Dept of Mathematics Research Papers

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