2014/10/05 by Konstantinos T. Gkikas, Gkikas, Konstantinos T., Laurent Veron +2
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Advanced Mathematical Physics Problems #Nonlinear Partial Differential Equations #math.AP #msc:28A12 #msc:31A15 #msc:35H25 #msc:35J10. #msc:35J66
paper · pdf · doi:10.48550/arxiv.1410.1176
77 pages
arxiv created 2014/10/27 · arxiv updated 2014/10/28
We study the boundary behaviour of the of (E) -\Gd u-\myfrac\xk d2(x)u+g(u)=0, where 0<\xk <(1)/(4) and g is a continuous nonndecreasing function in a bounded convex domain of \BBRN. We first construct the Martin kernel associated to the the linear operator \CL\xk =-\Gd-(\xk )/(d2(x)) and give a general condition for solving equation (E) with any Radon measure \gm for boundary data. When g(u)=|u|q-1u we show the existence of a critical exponent qc=qc(N,\xk )>1: when 0<q<qc any measure is eligible for solving (E) with \gm for boundary data; if q≥ qc, a necessary and sufficient condition is expressed in terms of the absolute continuity of \gmwith respect to some Besov capacity. The same capacity characterizes the removable compact boundary sets. At end any positive solution (F) -\Gd u-(\xk )/(d2(x))u+|u|q-1u=0 with q>1 admits a boundary trace which is a positive outer regular Borel measure. When 1<q<qc we prove that to any positive outer regular Borel measure we can associate a positive solutions of (F) with this boundary trace.