Regularity of solutions for an integral system of Wolff type

By Chao Ma, Wenxiong CHEN *, Congming LI #. 2011

109 citations recorded by the source corpus.

Abstract We consider the fully nonlinear integral systems involving Wolff potentials: (1) { u ( x ) = W β , γ ( v q ) ( x ) , x ∈ R n , v ( x ) = W β , γ ( u p ) ( x ) , x ∈ R n ; where W β , γ ( f ) ( x ) = ∫ 0 ∞ [ ∫ B t ( x ) f ( y ) d y t n − β γ ] 1 γ − 1 d t t . This system includes many known systems as special cases, in particular, when β = α 2 and γ = 2 , system (1) reduces to (2) { u ( x ) = ∫ R n 1 | x − y | n − α v ( y ) q d y , x ∈ R n , v ( x ) = ∫ R n 1 | x − y | n − α u ( y ) p d y , x ∈ R n . The solutions ( u , v ) of (2) are critical points of the functional associated with the well-known Hardy–Littlewood–Sobolev inequality. We can show that (2) is equivalent to a system of semi-linear elliptic PDEs { ( − Δ ) α / 2 u = v q , in R n , ( − Δ ) α / 2 v = u p , in R n , which comprises the well-known Lane–Emden system and Yamabe equation. We obtain integrability and regularity for the positive solutions to systems (1) . A regularity lifting method by contracting operators is used in proving the integrability, and while deriving the Lipschitz continuity, a brand new idea – Lifting Regularity by Shrinking Operators is introduced. We hope to see many more applications of this new idea in lifting regularities of solutions for nonlinear problems.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/85f71c1f78c0226f58db0fb7cbc92312f6e96cc6

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In Applied Mathematics

1. Congming LI #
2. Wenxiong CHEN *
3. Yutian Lei
4. Li Ma
5. Fengbo Hang
6. Dezhong Chen
7. G. Lu
8. Xiaohui Yu

In Mathematical Physics

1. Wenxiong CHEN *
2. Congming LI #
3. Yutian Lei
4. L. Martinazzi
5. Taoufik Hmidi

Citations in the corpus

• Super Polyharmonic Property of Solutions for PDE Systems and Its Applications (2011, methodology): “There are many more applications of this method to integral systems, and the interested readers may see [10] [12] [15] [20] [21] [28] [33] [34] [40] [42] [45] [46] [47] [50] [53] [54] and the references therein.”
• Sharp criteria of Liouville type for some nonlinear systems (2013, methodology): “Afterward, Ma, Chen and Li [39] used the regularity lifting lemmas to obtain the optimal integrability and the Lipschitz continuity.”
• Liouville type theorems for integral equations and integral systems (2013, unknown): “in [9] and [10] and then used widely in [6,8,17–19,23–26].”
• Hardy–Littlewood–Sobolev and Stein–Weiss inequalities and integral systems on the Heisenberg group (2012, unknown): “We omit the proof here since it is easy and can be found in [31].”