Pointwise estimates of Brezis–Kamin type for solutions of sublinear elliptic equations. (November 2016)

• Record Type: Journal Article
• Authors: Cao, Dat T.; Verbitsky, Igor E.
• Is Part Of: Nonlinear analysis. Volume 146(2016)
• Page Start: 1
• Page End: 19
• Publication Date: 2016-11
• Subjects: 35B05 -- 35J92; Quasilinear equations -- Fractional Laplacian -- Wolff potentials; 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.2016.08.008
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: We study quasilinear elliptic equations of the type − Δ p u = σ u q in R n, where Δ p 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 R n . 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 .