# American Institute of Mathematical Sciences

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. [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. [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. [4] M. Ghergu and V. Radulescu, "Nonlinear PDEs: Mathematical Models in Biology, Chemistry and Population Genetics,", Springer Monographs in Mathematics, (2011). [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. [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. [7] F. Gladiali and M. Squassina, On explosive solutions for a class of quasi-linear elliptic equations,, Adv. Nonlinear Stud., (). [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. [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. [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. [11] J. Serrin, A symmetry problem in potential theory,, Arch. Rational Mech. Anal., 43 (1971), 304. doi: 10.1007/BF00250468. [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. [13] M. Squassina, Existence, multiplicity, perturbation, and concentration results for a class of quasi-linear elliptic problems,, Electron. J. Differential Equations, 7 (2006). [14] M. Squassina, Symmetry in variational principles and applications,, J. London Math. Soc., 85 (2012), 323. doi: 10.1112/jlms/jdr046.

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##### 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. [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. [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. [4] M. Ghergu and V. Radulescu, "Nonlinear PDEs: Mathematical Models in Biology, Chemistry and Population Genetics,", Springer Monographs in Mathematics, (2011). [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. [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. [7] F. Gladiali and M. Squassina, On explosive solutions for a class of quasi-linear elliptic equations,, Adv. Nonlinear Stud., (). [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. [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. [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. [11] J. Serrin, A symmetry problem in potential theory,, Arch. Rational Mech. Anal., 43 (1971), 304. doi: 10.1007/BF00250468. [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. [13] M. Squassina, Existence, multiplicity, perturbation, and concentration results for a class of quasi-linear elliptic problems,, Electron. J. Differential Equations, 7 (2006). [14] M. Squassina, Symmetry in variational principles and applications,, J. London Math. Soc., 85 (2012), 323. doi: 10.1112/jlms/jdr046.
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