# American Institute of Mathematical Sciences

December  2015, 35(12): 5869-5877. doi: 10.3934/dcds.2015.35.5869

## Symmetry of components, Liouville-type theorems and classification results for some nonlinear elliptic systems

 1 LAMFA, CNRS UMR 7352, Université de Picardie Jules Verne, 33 rue Saint-Leu, 80039 Amiens

Received  December 2013 Published  May 2015

We prove the symmetry of components and some Liouville-type theorems for, possibly sign changing, entire distributional solutions to a family of nonlinear elliptic systems encompassing models arising in Bose-Einstein condensation and in nonlinear optics. For these models we also provide precise classification results for non-negative solutions. The sharpness of our results is also discussed.
Citation: Alberto Farina. Symmetry of components, Liouville-type theorems and classification results for some nonlinear elliptic systems. Discrete & Continuous Dynamical Systems, 2015, 35 (12) : 5869-5877. doi: 10.3934/dcds.2015.35.5869
##### References:
 [1] G. Bianchi, Non-existence of positive solutions to semilinear equations on $\mathbbR^n$ or $\mathbbR^n_+$ through the method of moving planes, Comm. Partial Diff. Eqns., 22 (1997), 1671-1690. doi: 10.1080/03605309708821315.  Google Scholar [2] H. Brezis, Semilinear equations in $R^N$ without condition at infinity, Appl. Math. Optim., 12 (1984), 271-282. doi: 10.1007/BF01449045.  Google Scholar [3] L. A. Caffarelli, B. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. Pure Appl. Math., 42 (1989), 271-297. doi: 10.1002/cpa.3160420304.  Google Scholar [4] C. Cortàzar, M. Elgueta and P. Felmer, On a semilinear elliptic problem in $\mathbbR^N$ with a non-Lipschitzian non-linearity, Advances in Differential Equations, 1 (1996), 199-218.  Google Scholar [5] E. N. Dancer and Y. Du, Some remarks on Liouville type results for quasilinear elliptic equations, Proc. Amer. Math. Soc., 131 (2003), 1891-1899. doi: 10.1090/S0002-9939-02-06733-3.  Google Scholar [6] A. Farina, Liouville-type theorems for elliptic problems, in Handbook of Differential Equations, Stationary Partial Differential Equations, 4, Elsevier, 2007, 61-116. doi: 10.1016/S1874-5733(07)80005-2.  Google Scholar [7] D. J. Frantzeskakis, Dark solitons in atomic Bose-Einstein condensates: From theory to experiments, J. Phys. A, 43 (2010), 213001, 68pp. doi: 10.1088/1751-8113/43/21/213001.  Google Scholar [8] B. Gidas, W.-M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $R^n$, Adv. Math. Supp. Stud., 7a (1981), 369-402.  Google Scholar [9] F. Gazzola, J. Serrin and M. Tang, Existence of ground states and free boundary problems for quasilinear elliptic operators, Advances in Differential Equations, 5 (2000), 1-30.  Google Scholar [10] B. Gidas and J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations, Comm. Partial Differential Equations, 6 (1981), 883-901. doi: 10.1080/03605308108820196.  Google Scholar [11] B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math., 34 (1981), 525-598. doi: 10.1002/cpa.3160340406.  Google Scholar [12] T. Kato, Schrödinger operators with singular potentials, Proceedings of the International Symposium on Partial Differential Equations and the Geometry of Normed Linear Spaces (Jerusalem, 1972), Israel J. Math., 13 (1972), 135-148. doi: 10.1007/BF02760233.  Google Scholar [13] J. B. Keller, On solutions of $\Delta u = f (u)$, Comm. Pure Appl. Math., 10 (1957), 503-510. doi: 10.1002/cpa.3160100402.  Google Scholar [14] Yu. S. Kivshar and B. Luther-Davies, Dark optical solitons: Physics and applications, Phys. Rep., 298 (1998), 81-197. doi: 10.1016/S0370-1573(97)00073-2.  Google Scholar [15] T.-C. Lin and J.-C. Wei, Symbiotic bright solitary wave solutions of coupled nonlinear Schrödinger equations, Nonlinearity, 19 (2006), 2755-2773. doi: 10.1088/0951-7715/19/12/002.  Google Scholar [16] L. Ma and L. Zhao, Uniqueness of ground states of some coupled nonlinear Schrödinger systems and their application, J. Differential Equations, 245 (2008), 2551-2565. doi: 10.1016/j.jde.2008.04.008.  Google Scholar [17] E. Mitidieri and S. I. Pohozaev, A priori estimates and blow-up of solutions of nonlinear partial differential equations and inequalities, Proc. Steklov Inst. Math., 234 (2001), 1-362.  Google Scholar [18] R. Osserman, On the inequality $\Delta u \ge f(u)$, Pacific J. Math., 7 (1957), 1641-1647.  Google Scholar [19] V. M. Perez-Garcia and J. B. Beitia, Symbiotic solitons in heteronuclear multicomponent Bose-Einstein condensates, Phys. Rev. A, 72 (2005), 033620. doi: 10.1103/PhysRevA.72.033620.  Google Scholar [20] P. Pucci and J. Serrin, Uniqueness of ground states for quasilinear elliptic operators, Indiana Univ. Math. J., 47 (1998), 501-528. doi: 10.1512/iumj.1998.47.1517.  Google Scholar [21] P. Quittner and P. Souplet, Symmetry of components for semilinear elliptic systems, SIAM J. Math. Anal., 44 (2012), 2545-2559. doi: 10.1137/11085428X.  Google Scholar [22] J. Serrin and M. Tang, Uniqueness of ground states for quasilinear elliptic equations, Indiana Univ. Math. J., 49 (2000), 897-923. doi: 10.1512/iumj.2000.49.1893.  Google Scholar [23] J. Serrin and H. Zou, Symmetry of ground states of quasilinear elliptic equations, Arch. Rational Mech. Anal., 148 (1999), 265-290. doi: 10.1007/s002050050162.  Google Scholar

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##### References:
 [1] G. Bianchi, Non-existence of positive solutions to semilinear equations on $\mathbbR^n$ or $\mathbbR^n_+$ through the method of moving planes, Comm. Partial Diff. Eqns., 22 (1997), 1671-1690. doi: 10.1080/03605309708821315.  Google Scholar [2] H. Brezis, Semilinear equations in $R^N$ without condition at infinity, Appl. Math. Optim., 12 (1984), 271-282. doi: 10.1007/BF01449045.  Google Scholar [3] L. A. Caffarelli, B. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. Pure Appl. Math., 42 (1989), 271-297. doi: 10.1002/cpa.3160420304.  Google Scholar [4] C. Cortàzar, M. Elgueta and P. Felmer, On a semilinear elliptic problem in $\mathbbR^N$ with a non-Lipschitzian non-linearity, Advances in Differential Equations, 1 (1996), 199-218.  Google Scholar [5] E. N. Dancer and Y. Du, Some remarks on Liouville type results for quasilinear elliptic equations, Proc. Amer. Math. Soc., 131 (2003), 1891-1899. doi: 10.1090/S0002-9939-02-06733-3.  Google Scholar [6] A. Farina, Liouville-type theorems for elliptic problems, in Handbook of Differential Equations, Stationary Partial Differential Equations, 4, Elsevier, 2007, 61-116. doi: 10.1016/S1874-5733(07)80005-2.  Google Scholar [7] D. J. Frantzeskakis, Dark solitons in atomic Bose-Einstein condensates: From theory to experiments, J. Phys. A, 43 (2010), 213001, 68pp. doi: 10.1088/1751-8113/43/21/213001.  Google Scholar [8] B. Gidas, W.-M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $R^n$, Adv. Math. Supp. Stud., 7a (1981), 369-402.  Google Scholar [9] F. Gazzola, J. Serrin and M. Tang, Existence of ground states and free boundary problems for quasilinear elliptic operators, Advances in Differential Equations, 5 (2000), 1-30.  Google Scholar [10] B. Gidas and J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations, Comm. Partial Differential Equations, 6 (1981), 883-901. doi: 10.1080/03605308108820196.  Google Scholar [11] B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math., 34 (1981), 525-598. doi: 10.1002/cpa.3160340406.  Google Scholar [12] T. Kato, Schrödinger operators with singular potentials, Proceedings of the International Symposium on Partial Differential Equations and the Geometry of Normed Linear Spaces (Jerusalem, 1972), Israel J. Math., 13 (1972), 135-148. doi: 10.1007/BF02760233.  Google Scholar [13] J. B. Keller, On solutions of $\Delta u = f (u)$, Comm. Pure Appl. Math., 10 (1957), 503-510. doi: 10.1002/cpa.3160100402.  Google Scholar [14] Yu. S. Kivshar and B. Luther-Davies, Dark optical solitons: Physics and applications, Phys. Rep., 298 (1998), 81-197. doi: 10.1016/S0370-1573(97)00073-2.  Google Scholar [15] T.-C. Lin and J.-C. Wei, Symbiotic bright solitary wave solutions of coupled nonlinear Schrödinger equations, Nonlinearity, 19 (2006), 2755-2773. doi: 10.1088/0951-7715/19/12/002.  Google Scholar [16] L. Ma and L. Zhao, Uniqueness of ground states of some coupled nonlinear Schrödinger systems and their application, J. Differential Equations, 245 (2008), 2551-2565. doi: 10.1016/j.jde.2008.04.008.  Google Scholar [17] E. Mitidieri and S. I. Pohozaev, A priori estimates and blow-up of solutions of nonlinear partial differential equations and inequalities, Proc. Steklov Inst. Math., 234 (2001), 1-362.  Google Scholar [18] R. Osserman, On the inequality $\Delta u \ge f(u)$, Pacific J. Math., 7 (1957), 1641-1647.  Google Scholar [19] V. M. Perez-Garcia and J. B. Beitia, Symbiotic solitons in heteronuclear multicomponent Bose-Einstein condensates, Phys. Rev. A, 72 (2005), 033620. doi: 10.1103/PhysRevA.72.033620.  Google Scholar [20] P. Pucci and J. Serrin, Uniqueness of ground states for quasilinear elliptic operators, Indiana Univ. Math. J., 47 (1998), 501-528. doi: 10.1512/iumj.1998.47.1517.  Google Scholar [21] P. Quittner and P. Souplet, Symmetry of components for semilinear elliptic systems, SIAM J. Math. Anal., 44 (2012), 2545-2559. doi: 10.1137/11085428X.  Google Scholar [22] J. Serrin and M. Tang, Uniqueness of ground states for quasilinear elliptic equations, Indiana Univ. Math. J., 49 (2000), 897-923. doi: 10.1512/iumj.2000.49.1893.  Google Scholar [23] J. Serrin and H. Zou, Symmetry of ground states of quasilinear elliptic equations, Arch. Rational Mech. Anal., 148 (1999), 265-290. doi: 10.1007/s002050050162.  Google Scholar
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