# 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 and 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), 69-85. 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), 1-18. 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), 1353-1385. 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, Springer Verlag, Heidelberg, xviii+392 pp., 2011. [5] B. Gidas, W.-M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68 (1979), 209-243. 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), 159-179. 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), 271-282. 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), 1753-1762. 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, Hindawi, 210 pp., 2008. doi: 10.1155/9789774540394. [11] J. Serrin, A symmetry problem in potential theory, Arch. Rational Mech. Anal., 43 (1971), 304-318. 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), 57-75. 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, Monograph 7, 2006, +213pp, Texas State University, USA. [14] M. Squassina, Symmetry in variational principles and applications, J. London Math. Soc., 85 (2012), 323-348. 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), 69-85. 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), 1-18. 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), 1353-1385. 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, Springer Verlag, Heidelberg, xviii+392 pp., 2011. [5] B. Gidas, W.-M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68 (1979), 209-243. 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), 159-179. 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), 271-282. 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), 1753-1762. 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, Hindawi, 210 pp., 2008. doi: 10.1155/9789774540394. [11] J. Serrin, A symmetry problem in potential theory, Arch. Rational Mech. Anal., 43 (1971), 304-318. 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), 57-75. 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, Monograph 7, 2006, +213pp, Texas State University, USA. [14] M. Squassina, Symmetry in variational principles and applications, J. London Math. Soc., 85 (2012), 323-348. doi: 10.1112/jlms/jdr046.
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