The precise boundary trace of positive solutions of the equation \Delta u = uq in the supercritical case

Moshe Marcus (1) , Laurent Veron (2)

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

We construct the precise boundary trace of positive solutions of $\Delta u=u^q$ in a smooth bounded domain $\Gw\sbs\BBR^N$, for $q$ in the super-critical case $q\geq (N+1)/(N-1)$. The construction is performed in the framework of the fine topology associated with the Bessel capacity $C_{{2/q,q'}}$ on $\bdw$. We prove that the boundary trace is a Borel measure (in general unbounded),which is outer regular and essentially absolutely continuous relative to this capacity. We provide a necessary and sufficient condition for such measures to be the boundary trace of a positive solution and prove that the corresponding generalized boundary value problem is well-posed in the class of $\sigma$-moderate solutions.

Mots clés: Nonlinear elliptic equations; Bessel capacities; fine topology; Borel measures.; Borel measures

Domaines: Equations aux dérivées partielles [math.AP]

Moshe Marcus, Laurent Veron. The precise boundary trace of positive solutions of the equation \Delta u = uq in the supercritical case. Contemporary mathematics, 2007, 446, pp.345-383. ⟨hal-00281807⟩