Multi‐Peak Solutions for a Wide Class of Singular Perturbation Problems
Multi‐Peak Solutions for a Wide Class of Singular Perturbation Problems
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DOI：
10.1112/s002461079900719x
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发表时间：
1999-04
期刊：
影响因子：
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通讯作者：
Juncheng Wei;M. Winter
中科院分区：
文献类型：
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作者：
Juncheng Wei;M. Winter
本文讨论了从相变、趋化性、图案形成、种群动力学和化学反应理论等不同领域产生的一类广泛的奇摄动问题。在没有任何对称性假设的情况下，研究了有界域上相应的椭圆型方程。假定边界的平均曲率有M个孤立的非退化临界点。然后证明了对于任意正整数M⩽M‘，存在一个稳定解，其中M个局部峰值位于边界上，且位于这些临界点附近.该方法基于Lyapunov-Schmidt约化。
This paper concerns a wide class of singular perturbation problems arising from such diverse fields as phase transitions, chemotaxis, pattern formation, population dynamics and chemical reaction theory. The corresponding elliptic equations in a bounded domain without any symmetry assumptions are studied. It is assumed that the mean curvature of the boundary has M¯ isolated, non‐degenerate critical points. Then it is shown that for any positive integer M⩽M¯ there exists a stationary solution with M local peaks which are attained on the boundary and which lie close to these critical points. The method is based on Lyapunov–Schmidt reduction.