The boundary trace and generalized boundary value problem for semilinear elliptic equations with coercive absorption
The boundary trace and generalized boundary value problem for semilinear elliptic equations with coercive absorption
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DOI：
10.1002/cpa.3037
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发表时间：
2003-06
影响因子：
3
通讯作者：
M. Marcus;L. Véron
中科院分区：
文献类型：
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作者：
M. Marcus;L. Véron
本文的第一部分建立了方程−Δu + g（x，u）= 0在光滑区域Ω N上的正解的边界迹的存在性。这个类包括每个满足Keller-Osserman条件的空间独立的单调递增g以及形式为gα，q（x，u）= d（x，<$Ω）α的退化非线性gα，q| u| q−1 u，其中α > −2且q > 1。边界迹由一个正的正则Borel测度给出，该测度在紧集上可能爆破。在第二部分中，我们集中讨论非线性项族{gα，q}，确定指数q的临界值（当α > −2时），并讨论（a）次临界非线性项具有孤立奇点的正解和（B）−Δu + gα，q（x，u）= 0的边值问题，边界数据由正则正Borel测度给出（可能无界）。我们表明，在亚临界的情况下，该问题拥有一个唯一的解决方案，为每一个这样的措施。© 2003 Wiley Periodicals，Inc.
In the first part of the paper we establish the existence of a boundary trace for positive solutions of the equation −Δu + g(x, u) = 0 in a smooth domain Ω ⊂ ℝN, for a general class of positive nonlinearities. This class includes every space independent, monotone increasing g which satisfies the Keller‐Osserman condition as well as degenerate nonlinearities gα,q of the form gα,q (x, u) = d(x, ∂Ω)α |u|q−1 u, with α > −2 and q > 1. The boundary trace is given by a positive regular Borel measure which may blow up on compact sets. In the second part we concentrate on the family of nonlinearities {gα,q}, determine the critical value of the exponent q (for fixed α > −2) and discuss (a) positive solutions with an isolated singularity, for subcritical nonlinearities and (b) the boundary value problem for −Δu + gα,q (x, u) = 0 with boundary data given by a positive regular Borel measure (possibly unbounded). We show that, in the subcritical case, the problem possesses a unique solution for every such measure. © 2003 Wiley Periodicals, Inc.