Boundary trace of positive solutions of supercritical semilinear elliptic equations in dihedral domains

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 (E)\; $-\Delta u+|u|^{q-1}u=0$ in a dihedral domain $\Gw$, when $q>1$ is supercritical. The value of the critical exponent can take only a finite number of values depending on the geometry of $\Gw$. When $\gm$ is a bounded Borel measure in a k-wedge, we give necessary and sufficient conditions in order it be the boundary value of a solution of (E). We also give conditions which ensure that a boundary compact subset is removable. These conditions are expressed in terms of Bessel capacities $B_{s,q'}$ in $\BBR^{N-k}$ where $s$ depends on the characteristics of the wedge. This allows us to describe the boundary trace of a positive solution of (E)

Mots clés: Laplacian; Poisson potential; Martin kernel; Borel measures; Besov spaces; harmonic lifting; Bessel capacities.; Bessel capacities

Domaines: Equations aux dérivées partielles [math.AP]

Moshe Marcus, Laurent Veron. Boundary trace of positive solutions of supercritical semilinear elliptic equations in dihedral domains. 2013. ⟨hal-00709073v3⟩