Behavior around isolated singularity for integral inequalities with multiple Riesz potentials. Issue 7 (2nd July 2020)

• Record Type: Journal Article
• Authors: Ghergu, Marius
• Is Part Of: Complex variables and elliptic equations. Volume 65:Issue 7(2020)
• Page Start: 1170
• Page End: 1182
• Publication Date: 2020-07-02
• Subjects: V. Radulescu; Isolated singularity -- integral inequality -- Riesz potentials -- Choquard inequality; 35J70; Functions of complex variables -- Periodicals; Differential equations, Elliptic -- Periodicals; 515.905
• DOI: 10.1080/17476933.2018.1551889
• Languages: English
• ISSNs: 1747-6933

Abstract

ABSTRACT: We are interested in the asymptotic behavior around the origin for nonnegative functions f ∈ C ( R n ∖ { 0 } ) ∩ L 1 ( R n ), n ≥ 1, satisfying the double integral inequality 0 ≤ f ( x ) ≤ | x | a − n + ∫ R n f ( y ) d y | x − y | n − a p × | x | b − n + ∫ R n f ( y ) d y | x − y | n − b q in B 1 ( 0 ) ∖ { 0 }, where p, q >0 and 0< a, b < n . We next apply these results to the study of the double elliptic inequality 0 ≤ − Δ u ≤ ∫ B 1 ( 0 ) u ( y ) d y | x − y | n − α p × ∫ B 1 ( 0 ) u ( y ) d y | x − y | n − β q in B 2 ( 0 ) ∖ { 0 }, where 0 ≤ u ∈ C 2 ( B 2 ( 0 ) ∖ { 0 } ) ∩ L 1 ( B 1 ( 0 ) ), n ≥ 3, p, q > 0 and 0 < α, β < n .