Mathematics > Analysis of PDEs
[Submitted on 25 Aug 2010 (v1), last revised 26 Oct 2011 (this version, v2)]
Title:Boundary Trace of Positive Solutions of Semilinear Elliptic Equations in Lipschitz Domains: The Subcritical Case
View PDFAbstract:We study the generalized boundary value problem for nonnegative solutions of of $-\Delta u+g(u)=0$ in a bounded Lipschitz domain $\Omega$, when $g$ is continuous and nondecreasing. Using the harmonic measure of $\Omega$, we define a trace in the class of outer regular Borel measures. We amphasize the case where $g(u)=|u|^{q-1}u$, $q>1$. When $\Omega$ is (locally) a cone with vertex $y$, we prove sharp results of removability and characterization of singular behavior. In the general case, assuming that $\Omega$ possesses a tangent cone at every boundary point and $q$ is subcritical, we prove an existence and uniqueness result for positive solutions with arbitrary boundary trace.
Submission history
From: Laurent Veron [view email] [via CCSD proxy][v1] Wed, 25 Aug 2010 07:32:31 UTC (58 KB)
[v2] Wed, 26 Oct 2011 12:59:27 UTC (59 KB)
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