BOUNDARY TRACE OF POSITIVE SOLUTIONS OF SEMILINEAR ELLIPTIC EQUATIONS IN LIPSCHITZ DOMAINS: THE SUBCRITICAL CASE

Moshe Marcus (1) , Laurent Veron (2)

1 TECHNION - Department of Mathematics
2 LMPT - Laboratoire de Mathématiques et Physique Théorique

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.

Mots clés: Laplacian; Poisson potential; harmonic measure; singularities; Borel measures; Harnack inequalities.; Harnack inequalities

Domaines: Equations aux dérivées partielles [math.AP]

Moshe Marcus, Laurent Veron. BOUNDARY TRACE OF POSITIVE SOLUTIONS OF SEMILINEAR ELLIPTIC EQUATIONS IN LIPSCHITZ DOMAINS: THE SUBCRITICAL CASE. 2010. ⟨hal-00511277v1⟩