2016/01/21 by Dat Cao, Dat T. Cao, Igor E. Verbitsky
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Advanced Mathematical Physics Problems #Bounded function #Combinatorics #Fractional Laplacian #Laplace operator #Mathematical analysis #Mathematics #Nabla symbol #Nonlinear Partial Differential Equations #Omega #Physics #Pointwise #Pure mathematics #Quantum mechanics #Sublinear function #Type (biology) #math.AP #msc:35B05 #msc:35J92 #p-Laplacian
paper · pdf · doi:10.1016/j.na.2016.08.008
published as Nonlinear Anal. 146 (2016), 1-19 · 24 pages
arxiv created 2016/01/21 · openalex publication_date 2016/09/12 · arxiv updated 2020/11/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We study quasilinear elliptic equations of the type -Δpu=σ uq in ℝn, where Δp u=∇ ⋅(∇ u |∇ u|p-2) is the p-Laplacian (or a more general A-Laplace operator div A(x, ∇ u)), 0<q < p-1, and σ≥ 0 is an arbitrary locally integrable function or measure on ℝn. We obtain necessary and sufficient conditions for the existence of positive solutions (not necessarily bounded) which satisfy global pointwise estimates of Brezis-Kamin type given in terms of Wolff potentials. Similar problems with the fractional Laplacian (-Δ)α for 0<α<(n)/(2) are treated as well, including explicit estimates for radially symmetric σ. Our results are new even in the classical case p=2 and α=1.