Regularity of solutions for an integral system of Wolff type |
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Authors: | Chao Ma Wenxiong Chen Congming Li |
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Institution: | aDepartment of Mathematics, University of Colorado at Boulder, Boulder, CO 80309, United States;bDepartment of Mathematics, Yeshiva University, 500 W 185th Street, New York, NY 10033, United States |
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Abstract: | We consider the fully nonlinear integral systems involving Wolff potentials:(1) where This system includes many known systems as special cases, in particular, when and γ=2, system (1) reduces to(2) 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 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. |
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Keywords: | Fully nonlinear Wolff potentials Integrability Lipschitz continuity Regularity liftings Shrinking operators |
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