2014/10/05 by Konstantinos T. Gkikas, Konstantinos Gkikas, Gkikas, Konstantinos +3
Mathematics · #Advanced Harmonic Analysis Research #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #math.AP
paper · pdf · doi:10.48550/arxiv.1410.1201
arxiv created 2014/10/05 · openalex publication_date 2014/10/05 · arxiv updated 2014/10/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let Ω⊂\BBRN be a bounded C2 domain and \CL_\gk=-\Gd-(\gk)/(d2) the Hardy operator where d=\dist (.,\prt\Gw) and 0<\gk≤(1)/(4). Let \ga±=1±√(1-4\gk) be the two Hardy exponents, \gl_\gk the first eigenvalue of \CL_\gk with corresponding positive eigenfunction ϕ_\gk. If g is a continuous nondecreasing function satisfying ∫1^∞(g(s)+|g(-s)|)s-2(2N-2+\ga+)/(2N-4+\ga+)ds<∞, then for any Radon measures \gn∈ \GTMϕ_\gk(\Gw) and \gm∈ \GTM(\prt\Gw) there exists a unique weak solution to problem P\gn,\gm: \CL_\gk u+g(u)=\gn in \Gw, u=\gm on \prt\Gw. If g(r)=|r|q-1u (q>1) we prove that, in the subcritical range of q, a necessary and sufficient condition for solving P0,\gm with \gm>0 is that \gm is absolutely continuous with respect to the capacity associated to the Besov space B2-(2+\ga+)/(2q'),q'(\BBRN-1). We also characterize the boundary removable sets in terms of this capacity. In the subcritical range of q we classify the isolated singularities of positive solutions.