Very weak solutions with boundary singularities for semilinear elliptic Dirichlet problems in domains with conical corners
Very weak solutions with boundary singularities for semilinear elliptic Dirichlet problems in domains with conical corners
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DOI：
10.1016/j.jmaa.2008.06.008
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发表时间：
2009-04
影响因子：
1.3
通讯作者：
J. Horák;P. J. McKenna;W. Reichel
中科院分区：
文献类型：
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作者：
J. Horák;P. J. McKenna;W. Reichel
设Ω <$Rn是一个在0∈ <$Ω有锥角的有界Lipschitz域.我们证明了问题−Δu=upin Ω，u=0在Ω上至少存在两个正的无界很弱解，它们在0处有一个奇异性，对任意p稍大于广义Brezis-Turner指数p*.在一个平面多边形域的例子上，计算了存在性结果成立的p-区间的实际大小。变分的解决方案被发现作为显式构造的奇异的锥解的扰动。这种方法也使得有可能找到数值逼近的两个非常弱的解决方案，Ω以下的梯度流的一个适当的功能和使用山路算法。二维的例子。
Let Ω⊂Rnbe a bounded Lipschitz domain with a cone-like corner at 0∈∂Ω. We prove existence of at least two positive unbounded very weak solutions of the problem −Δu=upin Ω, u=0 on ∂Ω, which have a singularity at 0, for any p slightly bigger that the generalized Brezis–Turner exponent p*. On an example of a planar polygonal domain the actual size of the p-interval on which the existence result holds is computed. The solutions are found variationally as perturbations of explicitly constructed singular solutions in cones. This approach also makes it possible to find numerical approximations of the two very weak solutions on Ω following a gradient flow of a suitable functional and using the mountain pass algorithm. Two-dimensional examples are presented.