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On the family of non-topological solutions for the elliptic system arising from a product Abelian gauge field theory
Sectional symmetry of solutions of elliptic systems in cylindrical domains
1. | Dipartimento di Matematica, Università di Roma, "Tor Vergata" - Via della Ricerca Scientifica 1 - 00133 Roma, Italy |
2. | Dipartimento di Matematica, Università di Roma, "Sapienza" - P.le A. Moro 2 - 00185 Roma, Italy |
In this paper we prove a kind of rotational symmetry for solutions of semilinear elliptic systems in some bounded cylindrical domains. The symmetry theorems obtained hold for low-Morse index solutions whenever the nonlinearities satisfy some convexity assumptions. These results extend and improve those obtained in [
References:
[1] |
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. |
[2] |
H. Berestycki and L. Nirenberg,
On the method of moving planes and the sliding method, Bol. Soc. Bras. Mat. (N.S.), 22 (1991), 1-37.
doi: 10.1007/BF01244896. |
[3] |
H. Berestycki, L. Nirenberg and S. R. S. Varadhan,
The principal eigenvalue and maximum principle for second order elliptic operators in general domains, Comm.Pure Appl. Math., 47 (1994), 47-92.
doi: 10.1002/cpa.3160470105. |
[4] |
J. Busca and B. Sirakov,
Symmetry results for semilinear elliptic systems in the whole space, Journal Diff. Eq., 163 (2000), 41-56.
doi: 10.1006/jdeq.1999.3701. |
[5] |
J. Busca and B. Sirakov,
Harnack type estimates for nonlinear ellyptic systems and applications, Ann. Inst. H. Poincaré Anal. Non Lin., 21 (2004), 543-590.
doi: 10.1016/j.anihpc.2003.06.001. |
[6] |
L. Damascelli and F. Pacella,
Symmetry results for cooperative elliptic systems via linearization, SIAM J. Math. Anal., 45 (2013), 1003-1026.
doi: 10.1137/110853534. |
[7] |
L. Damascelli and F. Pacella,
Morse index and symmetry for elliptic problems with nonlinear mixed boundary conditions, Proc. Royal S. Edinburgh Sect. A, 149 (2019), 305-324.
doi: 10.1017/prm.2018.29. |
[8] |
L. Damascelli and F. Pacella, Morse index of solutions of nonlinear elliptic equations and applications, De Gruyter Series in Nonlinear Analysis and Applications 30, De Gruyter, Berlin/Boston, (2019).
doi: 10.1515/9783110538243. |
[9] |
L. Damascelli, F. Gladiali and F. Pacella,
A symmetry result for semilinear cooperative elliptic systems, Recent trends in nonlinear partial differential equations. II. Stationary problems, Contemp. Math., Amer. Math. Soc. Providence, RI, 595 (2013), 187-204.
doi: 10.1090/conm/595/11802. |
[10] |
L. Damascelli, F. Gladiali and F. Pacella,
Symmetry results for cooperative elliptic systems in unbounded domains, Indiana Univ. Math. J., 63 (2014), 615-649.
doi: 10.1512/iumj.2014.63.5255. |
[11] |
D. G. de Figueiredo,
Monotonicity and symmetry of solutions of elliptic systems in general domains, NoDEA Nonlinear Differential Equations Appl., 1 (1994), 119-123.
doi: 10.1007/BF01193947. |
[12] |
D. G. de Figueiredo,
Semilinear elliptic systems: Existence, multiplicity, symmetry of solutions, Handbook of Differential Equations: Stationary Partial Differential Equations, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 5 (2008), 1-48.
doi: 10.1016/S1874-5733(08)80008-3. |
[13] |
D. G. de Figueiredo and E. Mitidieri,
Maximum principles for linear elliptic systems, Rend. Inst. Mat. Univ. Trieste, 22 (1992), 36-66.
|
[14] |
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. |
[15] |
F. Gladiali, F. Pacella and T. Weth,
Symmetry and Nonexistence of low Morse index solutions in unbounded domains, J. Math. Pures Appl., 93 (2010), 536-558.
doi: 10.1016/j.matpur.2009.08.003. |
[16] |
F. Pacella,
Symmetry of solutions to semilinear elliptic equations with convex nonlinearities, J. Funct. Anal., 192 (2002), 271-282.
doi: 10.1006/jfan.2001.3901. |
[17] |
F. Pacella and M. Ramaswamy,
Symmetry of solutions of elliptic equations via maximum principle, Handbook of Differential Equations: Stationary Partial Differential Equations, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 6 (2008), 269-312.
doi: 10.1016/S1874-5733(08)80021-6. |
[18] |
F. Pacella and T. Weth,
Symmetry of solutions to semilinear elliptic equations via Morse index, Proc. Amer. Math., 135 (2007), 1753-1762.
doi: 10.1090/S0002-9939-07-08652-2. |
[19] |
M. H. Protter and H. F. Weinberger, Maximum Principle in Differential Equations, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1967. |
[20] |
B. Sirakov,
Some estimates and maximum principles for weakly coupled systems of elliptic PDE, Nonlinear Analysis, 70 (2009), 3039-3046.
doi: 10.1016/j.na.2008.12.026. |
[21] |
D. Smets and M. Willem,
Partial symmetry and asimptotic behaviour for some elliptic variational problem, Calc. Var. Part. Diff. Eq., 18 (2003), 57-75.
doi: 10.1007/s00526-002-0180-y. |
[22] |
W. C. Troy,
Symmetry properties in systems of semilinear elliptic equations, J. Differential Eq., 42 (1981), 400-413.
doi: 10.1016/0022-0396(81)90113-3. |
[23] |
T. Weth,
Symmetry of solutions to variational problems for nonlinear elliptic equations via reflection methods, Jahresber. Dtsch. Math.-Ver., 112 (2010), 119-158.
doi: 10.1365/s13291-010-0005-4. |
show all references
References:
[1] |
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. |
[2] |
H. Berestycki and L. Nirenberg,
On the method of moving planes and the sliding method, Bol. Soc. Bras. Mat. (N.S.), 22 (1991), 1-37.
doi: 10.1007/BF01244896. |
[3] |
H. Berestycki, L. Nirenberg and S. R. S. Varadhan,
The principal eigenvalue and maximum principle for second order elliptic operators in general domains, Comm.Pure Appl. Math., 47 (1994), 47-92.
doi: 10.1002/cpa.3160470105. |
[4] |
J. Busca and B. Sirakov,
Symmetry results for semilinear elliptic systems in the whole space, Journal Diff. Eq., 163 (2000), 41-56.
doi: 10.1006/jdeq.1999.3701. |
[5] |
J. Busca and B. Sirakov,
Harnack type estimates for nonlinear ellyptic systems and applications, Ann. Inst. H. Poincaré Anal. Non Lin., 21 (2004), 543-590.
doi: 10.1016/j.anihpc.2003.06.001. |
[6] |
L. Damascelli and F. Pacella,
Symmetry results for cooperative elliptic systems via linearization, SIAM J. Math. Anal., 45 (2013), 1003-1026.
doi: 10.1137/110853534. |
[7] |
L. Damascelli and F. Pacella,
Morse index and symmetry for elliptic problems with nonlinear mixed boundary conditions, Proc. Royal S. Edinburgh Sect. A, 149 (2019), 305-324.
doi: 10.1017/prm.2018.29. |
[8] |
L. Damascelli and F. Pacella, Morse index of solutions of nonlinear elliptic equations and applications, De Gruyter Series in Nonlinear Analysis and Applications 30, De Gruyter, Berlin/Boston, (2019).
doi: 10.1515/9783110538243. |
[9] |
L. Damascelli, F. Gladiali and F. Pacella,
A symmetry result for semilinear cooperative elliptic systems, Recent trends in nonlinear partial differential equations. II. Stationary problems, Contemp. Math., Amer. Math. Soc. Providence, RI, 595 (2013), 187-204.
doi: 10.1090/conm/595/11802. |
[10] |
L. Damascelli, F. Gladiali and F. Pacella,
Symmetry results for cooperative elliptic systems in unbounded domains, Indiana Univ. Math. J., 63 (2014), 615-649.
doi: 10.1512/iumj.2014.63.5255. |
[11] |
D. G. de Figueiredo,
Monotonicity and symmetry of solutions of elliptic systems in general domains, NoDEA Nonlinear Differential Equations Appl., 1 (1994), 119-123.
doi: 10.1007/BF01193947. |
[12] |
D. G. de Figueiredo,
Semilinear elliptic systems: Existence, multiplicity, symmetry of solutions, Handbook of Differential Equations: Stationary Partial Differential Equations, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 5 (2008), 1-48.
doi: 10.1016/S1874-5733(08)80008-3. |
[13] |
D. G. de Figueiredo and E. Mitidieri,
Maximum principles for linear elliptic systems, Rend. Inst. Mat. Univ. Trieste, 22 (1992), 36-66.
|
[14] |
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. |
[15] |
F. Gladiali, F. Pacella and T. Weth,
Symmetry and Nonexistence of low Morse index solutions in unbounded domains, J. Math. Pures Appl., 93 (2010), 536-558.
doi: 10.1016/j.matpur.2009.08.003. |
[16] |
F. Pacella,
Symmetry of solutions to semilinear elliptic equations with convex nonlinearities, J. Funct. Anal., 192 (2002), 271-282.
doi: 10.1006/jfan.2001.3901. |
[17] |
F. Pacella and M. Ramaswamy,
Symmetry of solutions of elliptic equations via maximum principle, Handbook of Differential Equations: Stationary Partial Differential Equations, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 6 (2008), 269-312.
doi: 10.1016/S1874-5733(08)80021-6. |
[18] |
F. Pacella and T. Weth,
Symmetry of solutions to semilinear elliptic equations via Morse index, Proc. Amer. Math., 135 (2007), 1753-1762.
doi: 10.1090/S0002-9939-07-08652-2. |
[19] |
M. H. Protter and H. F. Weinberger, Maximum Principle in Differential Equations, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1967. |
[20] |
B. Sirakov,
Some estimates and maximum principles for weakly coupled systems of elliptic PDE, Nonlinear Analysis, 70 (2009), 3039-3046.
doi: 10.1016/j.na.2008.12.026. |
[21] |
D. Smets and M. Willem,
Partial symmetry and asimptotic behaviour for some elliptic variational problem, Calc. Var. Part. Diff. Eq., 18 (2003), 57-75.
doi: 10.1007/s00526-002-0180-y. |
[22] |
W. C. Troy,
Symmetry properties in systems of semilinear elliptic equations, J. Differential Eq., 42 (1981), 400-413.
doi: 10.1016/0022-0396(81)90113-3. |
[23] |
T. Weth,
Symmetry of solutions to variational problems for nonlinear elliptic equations via reflection methods, Jahresber. Dtsch. Math.-Ver., 112 (2010), 119-158.
doi: 10.1365/s13291-010-0005-4. |
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