Asymptotic symmetry and local behavior of solutions of higher order conformally invariant equations with isolated singularities |
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| Authors: | Tianling Jin Jingang Xiong |
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| Institution: | 1. Department of Mathematics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong;2. School of Mathematical Sciences, Laboratory of Mathematics and Complex Systems, MOE, Beijing Normal University, Beijing 100875, China;1. Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy;2. Dipartimento di Matematica, Università di Pisa, Largo B. Pontecorvo 5, 56217 Pisa, Italy;1. Instituto de Matemática, Universidade Federal do Rio de Janeiro, C. P. 68.530, CEP 21.945-970, Rio de Janeiro, RJ, Brazil;2. Department of Mathematics, University of Oklahoma, Norman, OK, USA;3. Department of Mathematics, Southern University of Science and Technology of China, Guangdong, China;4. Departamento de Geometria, Instituto de Matemática e Estatística, Universidade Federal Fluminense, Niterói, Brazil |
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| Abstract: | We prove sharp blow up rates of solutions of higher order conformally invariant equations in a bounded domain with an isolated singularity, and show the asymptotic radial symmetry of the solutions near the singularity. This is an extension of the celebrated theorem of Caffarelli-Gidas-Spruck for the second order Yamabe equation with isolated singularities to higher order equations. Our approach uses blow up analysis for local integral equations, and is unified for all critical elliptic equations of order smaller than the dimension. We also prove the existence of Fowler solutions to the global equations, and establish a sup ? inf type Harnack inequality of Schoen for integral equations. |
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| Keywords: | Conformal invariance Nonlocal equations Isolated singularity Asymptotic symmetry |
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