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