November  2013, 12(6): 3013-3026. doi: 10.3934/cpaa.2013.12.3013

On symmetry results for elliptic equations with convex nonlinearities

1. 

Department of Mathematical Sciences, Florida Institute of Technology, 150 W University Blvd, Melbourne, FL 32901

2. 

Dipartimento di Informatica, Università degli Studi di Verona, Cá Vignal 2, Strada Le Grazie 15, I-37134 Veron

Received  October 2012 Revised  March 2013 Published  May 2013

We investigate partial symmetry of solutions to semi-linear and quasi-linear elliptic problems with convex nonlinearities, in domains that are either axially symmetric or radially symmetric. The semi-linear problems are studied in a framework where the associated functional is of class $C^1$ but not of class $C^2$.
Citation: Kanishka Perera, Marco Squassina. On symmetry results for elliptic equations with convex nonlinearities. Communications on Pure & Applied Analysis, 2013, 12 (6) : 3013-3026. doi: 10.3934/cpaa.2013.12.3013
References:
[1]

T. Bartsch and M. Degiovanni, Nodal solutions of nonlinear elliptic Dirichlet problems on radial domains,, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei Mat. Appl., 17 (2006), 69.  doi: 10.4171/RLM/454.  Google Scholar

[2]

T. Bartsch, T. Weth and M. Willem, Partial symmetry of least energy nodal solutions to some variational problems,, J. Anal. Math., 96 (2005), 1.  doi: 10.1007/BF02787822.  Google Scholar

[3]

M. Colin, L. Jeanjean and M. Squassina, Stability and instability results for standing waves of quasi-linear Schrödinger equations,, Nonlinearity, 23 (2010), 1353.  doi: 10.1088/0951-7715/23/6/006.  Google Scholar

[4]

M. Ghergu and V. Radulescu, "Nonlinear PDEs: Mathematical Models in Biology, Chemistry and Population Genetics,", Springer Monographs in Mathematics, (2011).   Google Scholar

[5]

B. Gidas, W.-M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle,, Comm. Math. Phys., 68 (1979), 209.  doi: 10.1007/BF01221125.  Google Scholar

[6]

F. Gladiali and M. Squassina, Uniqueness of ground states for a class of quasi-linear elliptic equations,, Adv. Nonlinear Anal., 1 (2012), 159.  doi: 10.1515/ana-2011-0001.  Google Scholar

[7]

F. Gladiali and M. Squassina, On explosive solutions for a class of quasi-linear elliptic equations,, Adv. Nonlinear Stud., ().   Google Scholar

[8]

F. Pacella, Symmetry results for solutions of semilinear elliptic equations with convex non-linearities,, J. Funct. Anal., 192 (2002), 271.  doi: 10.1006/jfan.2001.3901.  Google Scholar

[9]

F. Pacella and T. Weth, Symmetry of solutions to semilinear elliptic equations via Morse index,, Proc. Amer. Math. Soc., 135 (2007), 1753.  doi: 10.1090/S0002-9939-07-08652-2.  Google Scholar

[10]

V. Radulescu, "Qualitative Analysis of Nonlinear Elliptic Partial Differential Equations: Monotonicity, Analytic, and Variational Methods,", Contemporary Mathematics and Its Applications, 6 (2008).  doi: 10.1155/9789774540394.  Google Scholar

[11]

J. Serrin, A symmetry problem in potential theory,, Arch. Rational Mech. Anal., 43 (1971), 304.  doi: 10.1007/BF00250468.  Google Scholar

[12]

D. Smets and M. Willem, Partial symmetry and asymptotic behaviour for some elliptic variational problems,, Calc. Var. Partial Differential Equations, 18 (2003), 57.  doi: 10.1007/s00526-002-0180-y.  Google Scholar

[13]

M. Squassina, Existence, multiplicity, perturbation, and concentration results for a class of quasi-linear elliptic problems,, Electron. J. Differential Equations, 7 (2006).   Google Scholar

[14]

M. Squassina, Symmetry in variational principles and applications,, J. London Math. Soc., 85 (2012), 323.  doi: 10.1112/jlms/jdr046.  Google Scholar

show all references

References:
[1]

T. Bartsch and M. Degiovanni, Nodal solutions of nonlinear elliptic Dirichlet problems on radial domains,, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei Mat. Appl., 17 (2006), 69.  doi: 10.4171/RLM/454.  Google Scholar

[2]

T. Bartsch, T. Weth and M. Willem, Partial symmetry of least energy nodal solutions to some variational problems,, J. Anal. Math., 96 (2005), 1.  doi: 10.1007/BF02787822.  Google Scholar

[3]

M. Colin, L. Jeanjean and M. Squassina, Stability and instability results for standing waves of quasi-linear Schrödinger equations,, Nonlinearity, 23 (2010), 1353.  doi: 10.1088/0951-7715/23/6/006.  Google Scholar

[4]

M. Ghergu and V. Radulescu, "Nonlinear PDEs: Mathematical Models in Biology, Chemistry and Population Genetics,", Springer Monographs in Mathematics, (2011).   Google Scholar

[5]

B. Gidas, W.-M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle,, Comm. Math. Phys., 68 (1979), 209.  doi: 10.1007/BF01221125.  Google Scholar

[6]

F. Gladiali and M. Squassina, Uniqueness of ground states for a class of quasi-linear elliptic equations,, Adv. Nonlinear Anal., 1 (2012), 159.  doi: 10.1515/ana-2011-0001.  Google Scholar

[7]

F. Gladiali and M. Squassina, On explosive solutions for a class of quasi-linear elliptic equations,, Adv. Nonlinear Stud., ().   Google Scholar

[8]

F. Pacella, Symmetry results for solutions of semilinear elliptic equations with convex non-linearities,, J. Funct. Anal., 192 (2002), 271.  doi: 10.1006/jfan.2001.3901.  Google Scholar

[9]

F. Pacella and T. Weth, Symmetry of solutions to semilinear elliptic equations via Morse index,, Proc. Amer. Math. Soc., 135 (2007), 1753.  doi: 10.1090/S0002-9939-07-08652-2.  Google Scholar

[10]

V. Radulescu, "Qualitative Analysis of Nonlinear Elliptic Partial Differential Equations: Monotonicity, Analytic, and Variational Methods,", Contemporary Mathematics and Its Applications, 6 (2008).  doi: 10.1155/9789774540394.  Google Scholar

[11]

J. Serrin, A symmetry problem in potential theory,, Arch. Rational Mech. Anal., 43 (1971), 304.  doi: 10.1007/BF00250468.  Google Scholar

[12]

D. Smets and M. Willem, Partial symmetry and asymptotic behaviour for some elliptic variational problems,, Calc. Var. Partial Differential Equations, 18 (2003), 57.  doi: 10.1007/s00526-002-0180-y.  Google Scholar

[13]

M. Squassina, Existence, multiplicity, perturbation, and concentration results for a class of quasi-linear elliptic problems,, Electron. J. Differential Equations, 7 (2006).   Google Scholar

[14]

M. Squassina, Symmetry in variational principles and applications,, J. London Math. Soc., 85 (2012), 323.  doi: 10.1112/jlms/jdr046.  Google Scholar

[1]

Qianqian Hou, Tai-Chia Lin, Zhi-An Wang. On a singularly perturbed semi-linear problem with Robin boundary conditions. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 401-414. doi: 10.3934/dcdsb.2020083

[2]

Alain Bensoussan, Xinwei Feng, Jianhui Huang. Linear-quadratic-Gaussian mean-field-game with partial observation and common noise. Mathematical Control & Related Fields, 2021, 11 (1) : 23-46. doi: 10.3934/mcrf.2020025

[3]

Shumin Li, Masahiro Yamamoto, Bernadette Miara. A Carleman estimate for the linear shallow shell equation and an inverse source problem. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 367-380. doi: 10.3934/dcds.2009.23.367

[4]

Lucio Damascelli, Filomena Pacella. Sectional symmetry of solutions of elliptic systems in cylindrical domains. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3305-3325. doi: 10.3934/dcds.2020045

[5]

Anh Tuan Duong, Phuong Le, Nhu Thang Nguyen. Symmetry and nonexistence results for a fractional Choquard equation with weights. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 489-505. doi: 10.3934/dcds.2020265

[6]

Denis Serre. Non-linear electromagnetism and special relativity. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 435-454. doi: 10.3934/dcds.2009.23.435

[7]

Vito Napolitano, Ferdinando Zullo. Codes with few weights arising from linear sets. Advances in Mathematics of Communications, 2020  doi: 10.3934/amc.2020129

[8]

Dandan Wang, Xiwang Cao, Gaojun Luo. A class of linear codes and their complete weight enumerators. Advances in Mathematics of Communications, 2021, 15 (1) : 73-97. doi: 10.3934/amc.2020044

[9]

Lihong Zhang, Wenwen Hou, Bashir Ahmad, Guotao Wang. Radial symmetry for logarithmic Choquard equation involving a generalized tempered fractional $ p $-Laplacian. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020445

[10]

Awais Younus, Zoubia Dastgeer, Nudrat Ishaq, Abdul Ghaffar, Kottakkaran Sooppy Nisar, Devendra Kumar. On the observability of conformable linear time-invariant control systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020444

[11]

Touria Karite, Ali Boutoulout. Global and regional constrained controllability for distributed parabolic linear systems: RHUM approach. Numerical Algebra, Control & Optimization, 2020  doi: 10.3934/naco.2020055

[12]

Noufel Frikha, Valentin Konakov, Stéphane Menozzi. Well-posedness of some non-linear stable driven SDEs. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 849-898. doi: 10.3934/dcds.2020302

[13]

He Zhang, John Harlim, Xiantao Li. Estimating linear response statistics using orthogonal polynomials: An RKHS formulation. Foundations of Data Science, 2020, 2 (4) : 443-485. doi: 10.3934/fods.2020021

[14]

Shudi Yang, Xiangli Kong, Xueying Shi. Complete weight enumerators of a class of linear codes over finite fields. Advances in Mathematics of Communications, 2021, 15 (1) : 99-112. doi: 10.3934/amc.2020045

[15]

Pablo Neme, Jorge Oviedo. A note on the lattice structure for matching markets via linear programming. Journal of Dynamics & Games, 2020  doi: 10.3934/jdg.2021001

[16]

Jong Yoon Hyun, Boran Kim, Minwon Na. Construction of minimal linear codes from multi-variable functions. Advances in Mathematics of Communications, 2021, 15 (2) : 227-240. doi: 10.3934/amc.2020055

[17]

Christian Clason, Vu Huu Nhu, Arnd Rösch. Optimal control of a non-smooth quasilinear elliptic equation. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020052

[18]

Yongge Tian, Pengyang Xie. Simultaneous optimal predictions under two seemingly unrelated linear random-effects models. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020168

[19]

Chao Wang, Qihuai Liu, Zhiguo Wang. Periodic bouncing solutions for Hill's type sub-linear oscillators with obstacles. Communications on Pure & Applied Analysis, 2021, 20 (1) : 281-300. doi: 10.3934/cpaa.2020266

[20]

Ali Mahmoodirad, Harish Garg, Sadegh Niroomand. Solving fuzzy linear fractional set covering problem by a goal programming based solution approach. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020162

2019 Impact Factor: 1.105

Metrics

  • PDF downloads (43)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]