On the existence of multipeaked solutions to a semilinear Neumann problem
On the existence of multipeaked solutions to a semilinear Neumann problem
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DOI：
10.1215/s0012-7094-99-09712-0
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发表时间：
1999-04
影响因子：
2.5
通讯作者：
D. Cao;T. Küpper
中科院分区：
文献类型：
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作者：
D. Cao;T. Küpper
∂ν = 0, x∈∂，其中是拉普拉斯算子，是R（N≥2）中的光滑有界域，ν是∂的单位向外法向量，20是常数。问题(P)e可以看作是生物学模式形成的原型。实际上，Keller和Segel将具有对数敏感性的趋化聚集模型的稳态问题简化为(P)e（见[20]）。此外，在Gierer和Meinhardt在[17]中提出的活化剂-抑制剂系统模拟生物模式形成的研究中，当抑制剂的扩散速率足够大时，(P)e起着重要作用。例如，请参阅[24]、[33]以及其中的参考资料了解更多细节。本文的动机之一是数值模拟（Keller-Segel模型以及Gierer-Meinhardt模型）中期望的(P)e解的“点凝聚现象”。也就是说，(P)e的解似乎在e−→0时趋向于零，除非在有限数量的点上。在这里，我们证明这些点被确定为边界∂的平均曲率的局部最大值和最小值点。结果由变分方法得到。定义“能量”
∂ν = 0, x ∈ ∂ , where is the Laplace operator, is a smooth bounded domain in R(N ≥ 2), ν is the unit outward normal vector to ∂ , 2 0 is a constant. Problem (P)e may be viewed as prototype of pattern formation in biology. Indeed, the steady-state problem for a chemotactic aggregation model with logarithmic sensitivity is reduced by Keller and Segel to (P)e (see [20]). Moreover, in the study of activator-inhibitor systems modeling biological pattern formation, proposed by Gierer and Meinhardt in [17], (P)e plays an important role when the diffusion rate of the inhibitor is sufficiently large. See, for example, [24], [33], and the references therein for more details. One of the motivations of this paper is the “point-condensation phenomena” of solutions to (P)e expected from numerical simulations (to the Keller-Segel model as well as to the Gierer-Meinhardt model). That is, the solutions to (P)e seem to tend to zero as e −→ 0 except at a finite number of points. Here we show that these points are determined as local maximun and minimum points of the mean curvature of the boundary ∂ . The results are obtained by variational methods. Define an “energy”