Boundary singularities of solutions of semilinear elliptic equations with critical Hardy potentials. (July 2015)

• Record Type: Journal Article
• Authors: Gkikas, Konstantinos T.; Véron, Laurent
• Is Part Of: Nonlinear analysis. Volume 121(2015)
• Page Start: 469
• Page End: 540
• Publication Date: 2015-07
• Subjects: primary 35J66 35J10 -- secondary 31A15 35H25 28A12; Semilinear elliptic equation -- Hardy potentials -- Harmonic measure -- Singular integrals -- Besov capacities -- Optimal lifting operator -- Boundary singularities -- Boundary trace; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2015.03.004
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: We study the boundary behavior of positive functions u satisfying (E) − Δ u − κ d 2 ( x ) u + g ( u ) = 0 in a bounded domain Ω of R N, where 0 < κ ≤ 1 4, g is a continuous nondecreasing function and d ( . ) is the distance function to ∂ Ω . We first construct the Martin kernel associated to the linear operator L κ = − Δ − κ d 2 ( x ) and give a general condition for solving equation (E) with any Radon measure μ for boundary data. When g ( u ) = | u | q − 1 u we show the existence of a critical exponent q c = q c ( N, κ ) > 1 with the following properties: when 0 < q < q c any measure is eligible for solving (E) with μ for boundary data; if q ≥ q c, a necessary and sufficient condition is expressed in terms of the absolute continuity of μ with respect to some Besov capacity. The same capacity characterizes the removable compact boundary sets. At end any positive solution (F) − Δ u − κ d 2 ( x ) u + | u | q − 1 u = 0 with q > 1 admits a boundary trace which is a positive outer regular Borel measure. When 1 < q < q c we prove that to any positive outer regular Borel measure we can associate a positive solutions of ( F ) with this boundary trace.