# American Institute of Mathematical Sciences

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September  2019, 18(5): 2819-2833. doi: 10.3934/cpaa.2019126

## A symmetry result for elliptic systems in punctured domains

 1 Dipartimento di Ingegneria Industriale e Scienze Matematiche, Università Politecnica della Marche, Via Brecce Bianche, 60131, Ancona, Italy 2 Department of Mathematics and Statistics, University of Western Australia, 35 Stirling Highway, WA 6009 Crawley, Australia 3 Dipartimento di Matematica, Università degli Studi di Trento, Via Sommarive 14, 38123, Povo (Trento), Italy

Received  October 2018 Revised  January 2019 Published  April 2019

Fund Project: The authors are members of INdAM/GNAMPA. The first and the third author are partially supported by the INdAM-GNAMPA Project 2018 "Problemi di curvatura relativi ad operatori ellittico-degeneri". The second author is supported by the Australian Research Council Discovery Project 170104880 NEW "Nonlocal Equations at Work"

We consider an elliptic system of equations in a punctured bounded domain. We prove that if the domain is convex in one direction and symmetric with respect to the reflections induced by the normal hyperplane to such a direction, then the solution is necessarily symmetric under this reflection and monotone in the corresponding direction. As a consequence, we prove symmetry results also for a related polyharmonic problem of any order with Navier boundary conditions.

Citation: Stefano Biagi, Enrico Valdinoci, Eugenio Vecchi. A symmetry result for elliptic systems in punctured domains. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2819-2833. doi: 10.3934/cpaa.2019126
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##### References:
 [1] Giuseppe Riey, Berardino Sciunzi. One dimensional symmetry of solutions to some anisotropic quasilinear elliptic equations in the plane. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1157-1166. doi: 10.3934/cpaa.2012.11.1157 [2] Serena Dipierro. Geometric inequalities and symmetry results for elliptic systems. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3473-3496. doi: 10.3934/dcds.2013.33.3473 [3] Claudianor O. Alves, Giovany M. Figueiredo, Marcelo F. Furtado. Multiplicity of solutions for elliptic systems via local Mountain Pass method. Communications on Pure & Applied Analysis, 2009, 8 (6) : 1745-1758. doi: 10.3934/cpaa.2009.8.1745 [4] Hwai-Chiuan Wang. Stability and symmetry breaking of solutions of semilinear elliptic equations. Conference Publications, 2005, 2005 (Special) : 886-894. doi: 10.3934/proc.2005.2005.886 [5] Pei Ma, Yan Li, Jihui Zhang. Symmetry and nonexistence of positive solutions for fractional systems. Communications on Pure & Applied Analysis, 2018, 17 (3) : 1053-1070. doi: 10.3934/cpaa.2018051 [6] Xavier Cabré. Elliptic PDE's in probability and geometry: Symmetry and regularity of solutions. Discrete & Continuous Dynamical Systems - A, 2008, 20 (3) : 425-457. doi: 10.3934/dcds.2008.20.425 [7] Shiren Zhu, Xiaoli Chen, Jianfu Yang. Regularity, symmetry and uniqueness of positive solutions to a nonlinear elliptic system. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2685-2696. doi: 10.3934/cpaa.2013.12.2685 [8] Sara Barile, Addolorata Salvatore. Radial solutions of semilinear elliptic equations with broken symmetry on unbounded domains. Conference Publications, 2013, 2013 (special) : 41-49. doi: 10.3934/proc.2013.2013.41 [9] Alberto Farina. Some symmetry results for entire solutions of an elliptic system arising in phase separation. Discrete & Continuous Dynamical Systems - A, 2014, 34 (6) : 2505-2511. doi: 10.3934/dcds.2014.34.2505 [10] Xianjin Chen, Jianxin Zhou. A local min-orthogonal method for multiple solutions of strongly coupled elliptic systems. Conference Publications, 2009, 2009 (Special) : 151-160. doi: 10.3934/proc.2009.2009.151 [11] Shuang Liu, Xinfeng Liu. Krylov implicit integration factor method for a class of stiff reaction-diffusion systems with moving boundaries. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-19. doi: 10.3934/dcdsb.2019176 [12] Wenxiong Chen, Congming Li. Radial symmetry of solutions for some integral systems of Wolff type. Discrete & Continuous Dynamical Systems - A, 2011, 30 (4) : 1083-1093. doi: 10.3934/dcds.2011.30.1083 [13] Ran Zhuo, Yan Li. Nonexistence and symmetry of solutions for Schrödinger systems involving fractional Laplacian. Discrete & Continuous Dynamical Systems - A, 2019, 39 (3) : 1595-1611. doi: 10.3934/dcds.2019071 [14] Alberto Farina. Symmetry of components, Liouville-type theorems and classification results for some nonlinear elliptic systems. Discrete & Continuous Dynamical Systems - A, 2015, 35 (12) : 5869-5877. doi: 10.3934/dcds.2015.35.5869 [15] Cristina Tarsi. Perturbation from symmetry and multiplicity of solutions for elliptic problems with subcritical exponential growth in $\mathbb{R} ^2$. Communications on Pure & Applied Analysis, 2008, 7 (2) : 445-456. doi: 10.3934/cpaa.2008.7.445 [16] Florin Catrina, Zhi-Qiang Wang. Asymptotic uniqueness and exact symmetry of k-bump solutions for a class of degenerate elliptic problems. Conference Publications, 2001, 2001 (Special) : 80-87. doi: 10.3934/proc.2001.2001.80 [17] Jian Hao, Zhilin Li, Sharon R. Lubkin. An augmented immersed interface method for moving structures with mass. Discrete & Continuous Dynamical Systems - B, 2012, 17 (4) : 1175-1184. doi: 10.3934/dcdsb.2012.17.1175 [18] Rumei Zhang, Jin Chen, Fukun Zhao. Multiple solutions for superlinear elliptic systems of Hamiltonian type. Discrete & Continuous Dynamical Systems - A, 2011, 30 (4) : 1249-1262. doi: 10.3934/dcds.2011.30.1249 [19] Weichung Wang, Tsung-Fang Wu, Chien-Hsiang Liu. On the multiple spike solutions for singularly perturbed elliptic systems. Discrete & Continuous Dynamical Systems - B, 2013, 18 (1) : 237-258. doi: 10.3934/dcdsb.2013.18.237 [20] Emmanuel Hebey, Jérôme Vétois. Multiple solutions for critical elliptic systems in potential form. Communications on Pure & Applied Analysis, 2008, 7 (3) : 715-741. doi: 10.3934/cpaa.2008.7.715

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