American Institute of Mathematical Sciences

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

On symmetry results for elliptic equations with convex nonlinearities

Kanishka Perera 1, and  Marco Squassina 2,

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 $C^2$.
Keywords: Semi-linear and quasi-linear elliptic equation, full and partial symmetry.
Mathematics Subject Classification: 35D99, 35J62, 58E05, 35J70.
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

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