Existence, local uniqueness and asymptotic approximation of spike solutions to singularly perturbed elliptic problems
Existence, local uniqueness and asymptotic approximation of spike solutions to singularly perturbed elliptic problems
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奇异摄动椭圆问题尖峰解的存在性、局部唯一性和渐近逼近
DOI：
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发表时间：
2015
期刊：
影响因子：
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通讯作者：
L. Recke
中科院分区：
文献类型：
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作者：
O. Omel’chenko;L. Recke
本文讨论一般奇摄动二阶半线性椭圆型方程在具有非线性自然边界条件的有界域上的解。这些方程不一定是变分类型的。我们描述了一种构造近似尖峰解序列的算法，证明了逼近近似尖峰解的精确解的存在和局部唯一性(使用隐函数定理类型的结果)，并估计了近似解与精确解之间的距离。在这里，尖峰解是指在W中存在一个点，使得该解在该点附近具有尖峰形状，并且该解在该点附近近似为零。穗的形状一般不是径向对称的，可能会改变符号。
This article is concerned with general singularly perturbed second order semilinear elliptic equations on bounded domains WHR with nonlinear natural boundary conditions. The equations are not necessarily of variational type. We describe an algorithm to construct sequences of approximate spike solutions, prove existence and local uniqueness of exact spike solutions close to the approximate ones (using an Implicit Function Theorem type result), and estimate the distance between the approximate and the exact solutions. Here spike solution means that there exists a point in W such that the solution has a spike-like shape in a vicinity of such point and that the solution is approximately zero away from this point. The spike shape is not radially symmetric in general and may change sign.