Boundary Trace of Positive Solutions of Semilinear Elliptic Equations in Lipschitz Domains: The Subcritical Case

Details Boundary Trace of Positive Solutions of Semilinear Elliptic Equations in Lipschitz Domains: The Subcritical Case Boundary Trace of Positive Solutions of Semilinear Elliptic Equations in Lipschitz Domains: The Subcritical Case

2010-08-25

arxiv.org/abs/1008.4222v2

Mathematics

Analysis of PDEs

To appear in Annali Scuola Normale Superiore Pisa. arXiv admin note: substantial text overlap with arXiv:0907.1006

Scientific paper

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.

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