Asymptotic behavior at isolated singularities for solutions of nonlocal semilinear elliptic systems of inequalities

We study the behavior near the origin of C 2 positive solutions u ( x ) and v ( x ) of the system 0 ≤ - Δ u ≤ 1 | x | α ∗ v λ 0 ≤ - Δ v ≤ 1 | x | β ∗ u σ in B 2 ( 0 ) \ { 0 } ⊂ R n , n ≥ 3 , where λ , σ ≥ 0 and α , β ∈ ( 0 , n ) . A by-product of our methods used to study these solutions will be res...

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Bibliographic Details
Published in:Calculus of variations and partial differential equations 2015-10, Vol.54 (2), p.1243-1273
Main Authors: Ghergu, Marius, Taliaferro, Steven D.
Format: Article
Language:English
Subjects:
Analysis
Calculus of Variations and Optimal Control
Optimization
Control
Mathematical and Computational Physics
Mathematics
Mathematics and Statistics
Systems Theory
Theoretical
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Summary:We study the behavior near the origin of C 2 positive solutions u ( x ) and v ( x ) of the system 0 ≤ - Δ u ≤ 1 | x | α ∗ v λ 0 ≤ - Δ v ≤ 1 | x | β ∗ u σ in B 2 ( 0 ) \ { 0 } ⊂ R n , n ≥ 3 , where λ , σ ≥ 0 and α , β ∈ ( 0 , n ) . A by-product of our methods used to study these solutions will be results on the behavior near the origin of L 1 ( B 1 ( 0 ) ) solutions f and g of the system 0 ≤ f ( x ) ≤ C | x | 2 - α + ∫ | y | < 1 g ( y ) d y | x - y | α - 2 λ 0 ≤ g ( x ) ≤ C | x | 2 - β + ∫ | y | < 1 f ( y ) d y | x - y | β - 2 σ for 0 < | x | < 1 where λ , σ ≥ 0 and α , β ∈ ( 2 , n + 2 ) .
ISSN:0944-2669
1432-0835
DOI:10.1007/s00526-015-0824-3