Sectional symmetry of solutions of elliptic systems in cylindrical domains

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June 2020, 40(6): 3305-3325. doi: 10.3934/dcds.2020045

Sectional symmetry of solutions of elliptic systems in cylindrical domains

Lucio Damascelli ^1, and Filomena Pacella ^2,

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

Received March 2019 Revised June 2019 Published October 2019

Fund Project: The first author acknowledges the MIUR Excellence Department Project awarded to the Department of Mathematics, University of Rome Tor Vergata, CUP E83C18000100006

Abstract
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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 [6,9,16,18].

Keywords: Foliated Schwarz symmetry, maximum principle, Morse index.

Mathematics Subject Classification: Primary: 3502, 35J61, 35P99; Secondary: 35B06, 35B07, 35B50.

Citation: Lucio Damascelli, Filomena Pacella. Sectional symmetry of solutions of elliptic systems in cylindrical domains. Discrete and Continuous Dynamical Systems, 2020, 40 (6) : 3305-3325. doi: 10.3934/dcds.2020045

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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