# American Institute of Mathematical Sciences

August  2013, 33(8): 3391-3405. doi: 10.3934/dcds.2013.33.3391

## Optimal partial regularity results for nonlinear elliptic systems in Carnot groups

 1 Department of Mathematics and Information Science, Zhangzhou Normal University, Zhangzhou 363000, Fujian, China 2 School of Mathematical Science, Xiamen University, Xiamen 361005, Fujian

Received  May 2012 Revised  July 2012 Published  January 2013

In this paper, we consider partial regularity for weak solutions of second-order nonlinear elliptic systems in Carnot groups. By the method of A-harmonic approximation, we establish optimal interior partial regularity of weak solutions to systems under controllable growth conditions with sub-quadratic growth in Carnot groups.
Citation: Shuhong Chen, Zhong Tan. Optimal partial regularity results for nonlinear elliptic systems in Carnot groups. Discrete & Continuous Dynamical Systems, 2013, 33 (8) : 3391-3405. doi: 10.3934/dcds.2013.33.3391
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##### References:
 [1] Shenzhou Zheng, Laping Zhang, Zhaosheng Feng. Everywhere regularity for P-harmonic type systems under the subcritical growth. Communications on Pure & Applied Analysis, 2008, 7 (1) : 107-117. doi: 10.3934/cpaa.2008.7.107 [2] Paulo Rabelo. Elliptic systems involving critical growth in dimension two. Communications on Pure & Applied Analysis, 2009, 8 (6) : 2013-2035. doi: 10.3934/cpaa.2009.8.2013 [3] Zhaoli Liu, Jiabao Su. Solutions of some nonlinear elliptic problems with perturbation terms of arbitrary growth. Discrete & Continuous Dynamical Systems, 2004, 10 (3) : 617-634. doi: 10.3934/dcds.2004.10.617 [4] Tian Xiang. Dynamics in a parabolic-elliptic chemotaxis system with growth source and nonlinear secretion. Communications on Pure & Applied Analysis, 2019, 18 (1) : 255-284. doi: 10.3934/cpaa.2019014 [5] Yilong Wang, Xuande Zhang. On a parabolic-elliptic chemotaxis-growth system with nonlinear diffusion. Discrete & Continuous Dynamical Systems - S, 2020, 13 (2) : 321-328. doi: 10.3934/dcdss.2020018 [6] Gabrielle Nornberg, Delia Schiera, Boyan Sirakov. A priori estimates and multiplicity for systems of elliptic PDE with natural gradient growth. Discrete & Continuous Dynamical Systems, 2020, 40 (6) : 3857-3881. doi: 10.3934/dcds.2020128 [7] João Marcos do Ó, Bruno Ribeiro, Bernhard Ruf. Hamiltonian elliptic systems in dimension two with arbitrary and double exponential growth conditions. Discrete & Continuous Dynamical Systems, 2021, 41 (1) : 277-296. doi: 10.3934/dcds.2020138 [8] Manassés de Souza. On a singular Hamiltonian elliptic systems involving critical growth in dimension two. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1859-1874. doi: 10.3934/cpaa.2012.11.1859 [9] Sami Aouaoui, Rahma Jlel. On some elliptic equation in the whole euclidean space $\mathbb{R}^2$ with nonlinearities having new exponential growth condition. Communications on Pure & Applied Analysis, 2020, 19 (10) : 4771-4796. doi: 10.3934/cpaa.2020211 [10] Claudianor O. Alves, César T. Ledesma. Multiplicity of solutions for a class of fractional elliptic problems with critical exponential growth and nonlocal Neumann condition. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021058 [11] VicenŢiu D. RǍdulescu, Somayeh Saiedinezhad. A nonlinear eigenvalue problem with $p(x)$-growth and generalized Robin boundary value condition. Communications on Pure & Applied Analysis, 2018, 17 (1) : 39-52. doi: 10.3934/cpaa.2018003 [12] Van Cyr, Bryna Kra. The automorphism group of a minimal shift of stretched exponential growth. Journal of Modern Dynamics, 2016, 10: 483-495. doi: 10.3934/jmd.2016.10.483 [13] Jun Wang, Junxiang Xu, Fubao Zhang. Homoclinic orbits for superlinear Hamiltonian systems without Ambrosetti-Rabinowitz growth condition. Discrete & Continuous Dynamical Systems, 2010, 27 (3) : 1241-1257. doi: 10.3934/dcds.2010.27.1241 [14] Oleksiy V. Kapustyan, Pavlo O. Kasyanov, José Valero. Regularity of global attractors for reaction-diffusion systems with no more than quadratic growth. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 1899-1908. doi: 10.3934/dcdsb.2017113 [15] Vicenţiu D. Rădulescu. Noncoercive elliptic equations with subcritical growth. Discrete & Continuous Dynamical Systems - S, 2012, 5 (4) : 857-864. doi: 10.3934/dcdss.2012.5.857 [16] Shenzhou Zheng, Xueliang Zheng, Zhaosheng Feng. Optimal regularity for $A$-harmonic type equations under the natural growth. Discrete & Continuous Dynamical Systems - B, 2011, 16 (2) : 669-685. doi: 10.3934/dcdsb.2011.16.669 [17] Casey Jao. Energy-critical NLS with potentials of quadratic growth. Discrete & Continuous Dynamical Systems, 2018, 38 (2) : 563-587. doi: 10.3934/dcds.2018025 [18] Luis Barreira, Claudia Valls. Quadratic Lyapunov sequences and arbitrary growth rates. Discrete & Continuous Dynamical Systems, 2010, 26 (1) : 63-74. doi: 10.3934/dcds.2010.26.63 [19] Giuliano Lazzaroni, Rodica Toader. Some remarks on the viscous approximation of crack growth. Discrete & Continuous Dynamical Systems - S, 2013, 6 (1) : 131-146. doi: 10.3934/dcdss.2013.6.131 [20] Sami Aouaoui. On some local-nonlocal elliptic equation involving nonlinear terms with exponential growth. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1767-1784. doi: 10.3934/cpaa.2017086

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