January  2020, 40(1): 549-577. doi: 10.3934/dcds.2020022

Symmetry and monotonicity properties of singular solutions to some cooperative semilinear elliptic systems involving critical nonlinearities

1. 

Università della Calabria, Dipartimento di Matematica e Informatica, Via P. Bucci 31B - Rende (CS), 87036, Italy

2. 

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

Received  April 2019 Revised  July 2019 Published  October 2019

We investigate qualitative properties of positive singular solutions of some elliptic systems in bounded and unbounded domains. We deduce symmetry and monotonicity properties via the moving plane procedure. Moreover, in the unbounded case, we study some cooperative elliptic systems involving critical nonlinearities in $ {\mathbb{R}}^n $.

Citation: Francesco Esposito. Symmetry and monotonicity properties of singular solutions to some cooperative semilinear elliptic systems involving critical nonlinearities. Discrete & Continuous Dynamical Systems - A, 2020, 40 (1) : 549-577. doi: 10.3934/dcds.2020022
References:
[1]

A. D. Alexandrov, A characteristic property of the spheres, Ann. Mat. Pura Appl., 58 (1962), 303-354.  doi: 10.1007/BF02413056.  Google Scholar

[2]

A. Ambrosetti and E. Colorado, Standing waves of some coupled nonlinear Schrödinger equations, J. Lond. Math. Soc., 75 (2007), 67-82.  doi: 10.1112/jlms/jdl020.  Google Scholar

[3]

T. BartschN. Dancer and Z. Q. Wang, A Liouville theorem, a-priori bounds, and bifurcating branches of positive solutions for a nonlinear elliptic system, Calc. Var. Partial Differ. Equ., 37 (2010), 345-361.  doi: 10.1007/s00526-009-0265-y.  Google Scholar

[4]

T. BartschZ. Q. Wang and J. Wei, Bound states for a coupled Schrödinger system, J. Fixed Point Theory Appl., 2 (2007), 353-367.  doi: 10.1007/s11784-007-0033-6.  Google Scholar

[5]

H. Berestycki and L. Nirenberg, On the method of moving planes and the sliding method, Bulletin Soc. Brasil. de Mat Nova Ser, 22 (1991), 1-37.  doi: 10.1007/BF01244896.  Google Scholar

[6]

S. BiagiE. Valdinoci and E. Vecchi, A symmetry result for elliptic systems in punctured domains, Commun. Pure Appl. Anal., 18 (2019), 2819-2833.  doi: 10.3934/cpaa.2019126.  Google Scholar

[7]

S. Biagi, E. Valdinoci and E. Vecchi, A symmetry result for cooperative elliptic systems with singularities, arXiv: 1904.02003. Google Scholar

[8]

L. Boccardo and L. Orsina, Semilinear elliptic equations with singular nonlinearities, Calc. Var. Partial Differential Equations, 37 (2010), 363-380.  doi: 10.1007/s00526-009-0266-x.  Google Scholar

[9]

H. Brezis and T. Kato, Remarks on the Schrödinger operator with singular complex potentials, J. Math. Pures Appl., 58 (1979), 137-151.   Google Scholar

[10]

J. Busca and B. Sirakov, Symmetry results for semilinear elliptic systems in the whole space, J. Differential Equations, 163 (2000), 41-56.  doi: 10.1006/jdeq.1999.3701.  Google Scholar

[11]

L. A. CaffarelliB. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. on Pure and Appl. Mat., 42 (1989), 271-297.  doi: 10.1002/cpa.3160420304.  Google Scholar

[12]

A. CaninoF. Esposito and B. Sciunzi, On the Höpf boundary lemma for singular semilinear elliptic equations, J. of Differential Equations, 266 (2019), 5488-5499.  doi: 10.1016/j.jde.2018.10.039.  Google Scholar

[13]

M. Clapp and A. Pistoia, Existence and phase separation of entire solutions to a pure critical competitive elliptic system, Calc. Var. Partial Differential Equations, 57 (2018), Art. 23, 20 pp. doi: 10.1007/s00526-017-1283-9.  Google Scholar

[14]

M. G. CrandallP. H. Rabinowitz and L. Tartar, On a Dirichlet problem with a singular nonlinearity, Comm. P.D.E., 2 (1977), 193-222.  doi: 10.1080/03605307708820029.  Google Scholar

[15]

E. N. Dancer, Moving plane methods for systems on half spaces, Math. Ann., 342 (2008), 245-254.  doi: 10.1007/s00208-008-0226-3.  Google Scholar

[16]

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.  Google Scholar

[17]

D. G. De Figueiredo and J. Yang, Decay, symmetry and existence of solutions of semilinear elliptic systems, Nonlinear Anal., 33 (1998), 211-234.  doi: 10.1016/S0362-546X(97)00548-8.  Google Scholar

[18]

E. De Giorgi, Un esempio di estremali discontinue per un problema variazionale di tipo ellittico, Boll. Un. Mat. Ital., 1 (1968), 135-137.   Google Scholar

[19]

F. EspositoA. Farina and B. Sciunzi, Qualitative properties of singular solutions to semilinear elliptic problems, J. of Differential Equations, 265 (2018), 1962-1983.  doi: 10.1016/j.jde.2018.04.030.  Google Scholar

[20]

F. EspositoL. Montoro and B. Sciunzi, Monotonicity and symmetry of singular solutions to quasilinear problems, J. Math. Pure Appl., 126 (2019), 214-231.  doi: 10.1016/j.matpur.2018.09.005.  Google Scholar

[21]

F. Esposito and B. Sciunzi, On the Höpf boundary lemma for quasilinear problems involving singular nonlinearities and applications, arXiv: 1810.13294. Google Scholar

[22]

L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, Studies in Advanced Mathematics, CRC Press, 1992.  Google Scholar

[23]

B. GidasW. M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68 (1979), 209-243.  doi: 10.1007/BF01221125.  Google Scholar

[24]

F. Gladiali, M. Grossi and C. Troesler, A non-variational system involving the critical Sobolev exponent. The radial case, J. Anal. Math., 138 (2019), 643–671, arXiv: 1603.05641. doi: 10.1007/s11854-019-0040-8.  Google Scholar

[25]

F. Gladiali, M. Grossi and C. Troesler, Entire radial and nonradial solutions for systems with critical growth, Calc. Var. Partial Differential Equations, 57 (2018), Art. 53, 26 pp. doi: 10.1007/s00526-018-1340-z.  Google Scholar

[26]

Y. Guo and J. Liu, Liouville type theorems for positive solutions of elliptic system in $ {\mathbb{R}}^n$, Comm. Partial Differential Equations, 33 (2008), 263-284.  doi: 10.1080/03605300701257476.  Google Scholar

[27]

A. C. Lazer and P. J. McKenna, On a singular nonlinear elliptic boundary-value problem, Proc. AMS, 111 (1991), 721-730.  doi: 10.1090/S0002-9939-1991-1037213-9.  Google Scholar

[28]

G. Leoni and M. Morini, Necessary and sufficient conditions for the chain rule in $W^{1, 1}_{loc}({\mathbb{R}}^n; {\mathbb{R}}^d)$ and $BV_{loc}({\mathbb{R}}^n; {\mathbb{R}}^d)$, J. Eur. Math. Soc., 9 (2007), 219-252.  doi: 10.4171/JEMS/78.  Google Scholar

[29]

T. C. Lin and J. Wei, Ground state of $N$ coupled nonlinear Schrödinger equations in $ {\mathbb{R}}^n$, $n \leq 3$, Commun. Math. Phys., 255 (2005), 629-653.  doi: 10.1007/s00220-005-1313-x.  Google Scholar

[30]

T. C. Lin and J. Wei, Spikes in two coupled nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2005), 403-439.  doi: 10.1016/j.anihpc.2004.03.004.  Google Scholar

[31]

E. Mitidieri, A Rellich type identity and applications, Comm. Partial Differential Equations, 18 (1993), 125-151.  doi: 10.1080/03605309308820923.  Google Scholar

[32]

E. Mitidieri, Nonexistence of positive solutions of semilinear elliptic systems in $ {\mathbb{R}}^n$, Differential Integral Equations, 9 (1996), 465-479.   Google Scholar

[33]

L. Montoro, F. Punzo and B. Sciunzi, Qualitative properties of singular solutions to nonlocal problems, Ann. Mat. Pura Appl. (4), 197 (2018), 941–964. doi: 10.1007/s10231-017-0710-z.  Google Scholar

[34]

L. MontoroG. Riey and B. Sciunzi, Qualitative properties of positive solutions to systems of quasilinear elliptic equations, Adv. Differential Equations, 20 (2015), 717-740.   Google Scholar

[35]

L. MontoroB. Sciunzi and M. Squassina, Symmetry results for nonvariational quasi-linear elliptic systems, Adv. Nonlinear Stud., 10 (2010), 939-955.  doi: 10.1515/ans-2010-0411.  Google Scholar

[36]

S. Peng, Y. F. Peng and Z. O. Wang, On elliptic systems with Sobolev critical growth, Calc. Var. Partial Differential Equations, 55 (2016), Art. 142, 30 pp. doi: 10.1007/s00526-016-1091-7.  Google Scholar

[37]

W. Reichel and H. Zou, Non-existence results for semilinear cooperative elliptic systems via moving spheres, J. Differential Equations, 161 (2000), 219-243.  doi: 10.1006/jdeq.1999.3700.  Google Scholar

[38]

B. Sciunzi, On the moving Plane Method for singular solutions to semilinear elliptic equations, J. Math. Pures Appl., 108 (2017), 111-123.  doi: 10.1016/j.matpur.2016.10.012.  Google Scholar

[39]

J. Serrin, A symmetry problem in potential theory, Arch. Rational Mech. Anal., 43 (1971), 304-318.  doi: 10.1007/BF00250468.  Google Scholar

[40]

B. Sirakov, Least energy solitary waves for a system of nonlinear Schrödinger equations in $ {\mathbb{R}}^n$, Commun. Math. Phys., 271 (2007), 199-221.  doi: 10.1007/s00220-006-0179-x.  Google Scholar

[41]

N. Soave, On existence and phase separation of solitary waves for nonlinear Schrödinger systems modelling simultaneous cooperation and competition, Calc. Var. Partial Differ. Equ., 53 (2015), 689-718.  doi: 10.1007/s00526-014-0764-3.  Google Scholar

[42]

N. Soave and H. Tavares, New existence and symmetry results for least energy positive solutions of Schrödinger systems with mixed competition and cooperation terms, J. Differential Equations, 261 (2016), 505-537.  doi: 10.1016/j.jde.2016.03.015.  Google Scholar

[43]

M. Struwe, Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems. Fourth Edition, Springer-Verlag, Berlin, 2008.  Google Scholar

[44]

C. A. Stuart, Existence and approximation of solutions of nonlinear elliptic equations, Math. Z., 147 (1976), 53-63.  doi: 10.1007/BF01214274.  Google Scholar

[45]

S. Terracini, On positive entire solutions to a class of equations with a singular coefficient and critical exponent, Adv. Differential Equations, 1 (1996), 241-264.   Google Scholar

[46]

W. C. Troy, Symmetry properties in systems of semilinear elliptic equations, J. of Differential Equations, 42 (1981), 400-413.  doi: 10.1016/0022-0396(81)90113-3.  Google Scholar

show all references

References:
[1]

A. D. Alexandrov, A characteristic property of the spheres, Ann. Mat. Pura Appl., 58 (1962), 303-354.  doi: 10.1007/BF02413056.  Google Scholar

[2]

A. Ambrosetti and E. Colorado, Standing waves of some coupled nonlinear Schrödinger equations, J. Lond. Math. Soc., 75 (2007), 67-82.  doi: 10.1112/jlms/jdl020.  Google Scholar

[3]

T. BartschN. Dancer and Z. Q. Wang, A Liouville theorem, a-priori bounds, and bifurcating branches of positive solutions for a nonlinear elliptic system, Calc. Var. Partial Differ. Equ., 37 (2010), 345-361.  doi: 10.1007/s00526-009-0265-y.  Google Scholar

[4]

T. BartschZ. Q. Wang and J. Wei, Bound states for a coupled Schrödinger system, J. Fixed Point Theory Appl., 2 (2007), 353-367.  doi: 10.1007/s11784-007-0033-6.  Google Scholar

[5]

H. Berestycki and L. Nirenberg, On the method of moving planes and the sliding method, Bulletin Soc. Brasil. de Mat Nova Ser, 22 (1991), 1-37.  doi: 10.1007/BF01244896.  Google Scholar

[6]

S. BiagiE. Valdinoci and E. Vecchi, A symmetry result for elliptic systems in punctured domains, Commun. Pure Appl. Anal., 18 (2019), 2819-2833.  doi: 10.3934/cpaa.2019126.  Google Scholar

[7]

S. Biagi, E. Valdinoci and E. Vecchi, A symmetry result for cooperative elliptic systems with singularities, arXiv: 1904.02003. Google Scholar

[8]

L. Boccardo and L. Orsina, Semilinear elliptic equations with singular nonlinearities, Calc. Var. Partial Differential Equations, 37 (2010), 363-380.  doi: 10.1007/s00526-009-0266-x.  Google Scholar

[9]

H. Brezis and T. Kato, Remarks on the Schrödinger operator with singular complex potentials, J. Math. Pures Appl., 58 (1979), 137-151.   Google Scholar

[10]

J. Busca and B. Sirakov, Symmetry results for semilinear elliptic systems in the whole space, J. Differential Equations, 163 (2000), 41-56.  doi: 10.1006/jdeq.1999.3701.  Google Scholar

[11]

L. A. CaffarelliB. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. on Pure and Appl. Mat., 42 (1989), 271-297.  doi: 10.1002/cpa.3160420304.  Google Scholar

[12]

A. CaninoF. Esposito and B. Sciunzi, On the Höpf boundary lemma for singular semilinear elliptic equations, J. of Differential Equations, 266 (2019), 5488-5499.  doi: 10.1016/j.jde.2018.10.039.  Google Scholar

[13]

M. Clapp and A. Pistoia, Existence and phase separation of entire solutions to a pure critical competitive elliptic system, Calc. Var. Partial Differential Equations, 57 (2018), Art. 23, 20 pp. doi: 10.1007/s00526-017-1283-9.  Google Scholar

[14]

M. G. CrandallP. H. Rabinowitz and L. Tartar, On a Dirichlet problem with a singular nonlinearity, Comm. P.D.E., 2 (1977), 193-222.  doi: 10.1080/03605307708820029.  Google Scholar

[15]

E. N. Dancer, Moving plane methods for systems on half spaces, Math. Ann., 342 (2008), 245-254.  doi: 10.1007/s00208-008-0226-3.  Google Scholar

[16]

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.  Google Scholar

[17]

D. G. De Figueiredo and J. Yang, Decay, symmetry and existence of solutions of semilinear elliptic systems, Nonlinear Anal., 33 (1998), 211-234.  doi: 10.1016/S0362-546X(97)00548-8.  Google Scholar

[18]

E. De Giorgi, Un esempio di estremali discontinue per un problema variazionale di tipo ellittico, Boll. Un. Mat. Ital., 1 (1968), 135-137.   Google Scholar

[19]

F. EspositoA. Farina and B. Sciunzi, Qualitative properties of singular solutions to semilinear elliptic problems, J. of Differential Equations, 265 (2018), 1962-1983.  doi: 10.1016/j.jde.2018.04.030.  Google Scholar

[20]

F. EspositoL. Montoro and B. Sciunzi, Monotonicity and symmetry of singular solutions to quasilinear problems, J. Math. Pure Appl., 126 (2019), 214-231.  doi: 10.1016/j.matpur.2018.09.005.  Google Scholar

[21]

F. Esposito and B. Sciunzi, On the Höpf boundary lemma for quasilinear problems involving singular nonlinearities and applications, arXiv: 1810.13294. Google Scholar

[22]

L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, Studies in Advanced Mathematics, CRC Press, 1992.  Google Scholar

[23]

B. GidasW. M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68 (1979), 209-243.  doi: 10.1007/BF01221125.  Google Scholar

[24]

F. Gladiali, M. Grossi and C. Troesler, A non-variational system involving the critical Sobolev exponent. The radial case, J. Anal. Math., 138 (2019), 643–671, arXiv: 1603.05641. doi: 10.1007/s11854-019-0040-8.  Google Scholar

[25]

F. Gladiali, M. Grossi and C. Troesler, Entire radial and nonradial solutions for systems with critical growth, Calc. Var. Partial Differential Equations, 57 (2018), Art. 53, 26 pp. doi: 10.1007/s00526-018-1340-z.  Google Scholar

[26]

Y. Guo and J. Liu, Liouville type theorems for positive solutions of elliptic system in $ {\mathbb{R}}^n$, Comm. Partial Differential Equations, 33 (2008), 263-284.  doi: 10.1080/03605300701257476.  Google Scholar

[27]

A. C. Lazer and P. J. McKenna, On a singular nonlinear elliptic boundary-value problem, Proc. AMS, 111 (1991), 721-730.  doi: 10.1090/S0002-9939-1991-1037213-9.  Google Scholar

[28]

G. Leoni and M. Morini, Necessary and sufficient conditions for the chain rule in $W^{1, 1}_{loc}({\mathbb{R}}^n; {\mathbb{R}}^d)$ and $BV_{loc}({\mathbb{R}}^n; {\mathbb{R}}^d)$, J. Eur. Math. Soc., 9 (2007), 219-252.  doi: 10.4171/JEMS/78.  Google Scholar

[29]

T. C. Lin and J. Wei, Ground state of $N$ coupled nonlinear Schrödinger equations in $ {\mathbb{R}}^n$, $n \leq 3$, Commun. Math. Phys., 255 (2005), 629-653.  doi: 10.1007/s00220-005-1313-x.  Google Scholar

[30]

T. C. Lin and J. Wei, Spikes in two coupled nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2005), 403-439.  doi: 10.1016/j.anihpc.2004.03.004.  Google Scholar

[31]

E. Mitidieri, A Rellich type identity and applications, Comm. Partial Differential Equations, 18 (1993), 125-151.  doi: 10.1080/03605309308820923.  Google Scholar

[32]

E. Mitidieri, Nonexistence of positive solutions of semilinear elliptic systems in $ {\mathbb{R}}^n$, Differential Integral Equations, 9 (1996), 465-479.   Google Scholar

[33]

L. Montoro, F. Punzo and B. Sciunzi, Qualitative properties of singular solutions to nonlocal problems, Ann. Mat. Pura Appl. (4), 197 (2018), 941–964. doi: 10.1007/s10231-017-0710-z.  Google Scholar

[34]

L. MontoroG. Riey and B. Sciunzi, Qualitative properties of positive solutions to systems of quasilinear elliptic equations, Adv. Differential Equations, 20 (2015), 717-740.   Google Scholar

[35]

L. MontoroB. Sciunzi and M. Squassina, Symmetry results for nonvariational quasi-linear elliptic systems, Adv. Nonlinear Stud., 10 (2010), 939-955.  doi: 10.1515/ans-2010-0411.  Google Scholar

[36]

S. Peng, Y. F. Peng and Z. O. Wang, On elliptic systems with Sobolev critical growth, Calc. Var. Partial Differential Equations, 55 (2016), Art. 142, 30 pp. doi: 10.1007/s00526-016-1091-7.  Google Scholar

[37]

W. Reichel and H. Zou, Non-existence results for semilinear cooperative elliptic systems via moving spheres, J. Differential Equations, 161 (2000), 219-243.  doi: 10.1006/jdeq.1999.3700.  Google Scholar

[38]

B. Sciunzi, On the moving Plane Method for singular solutions to semilinear elliptic equations, J. Math. Pures Appl., 108 (2017), 111-123.  doi: 10.1016/j.matpur.2016.10.012.  Google Scholar

[39]

J. Serrin, A symmetry problem in potential theory, Arch. Rational Mech. Anal., 43 (1971), 304-318.  doi: 10.1007/BF00250468.  Google Scholar

[40]

B. Sirakov, Least energy solitary waves for a system of nonlinear Schrödinger equations in $ {\mathbb{R}}^n$, Commun. Math. Phys., 271 (2007), 199-221.  doi: 10.1007/s00220-006-0179-x.  Google Scholar

[41]

N. Soave, On existence and phase separation of solitary waves for nonlinear Schrödinger systems modelling simultaneous cooperation and competition, Calc. Var. Partial Differ. Equ., 53 (2015), 689-718.  doi: 10.1007/s00526-014-0764-3.  Google Scholar

[42]

N. Soave and H. Tavares, New existence and symmetry results for least energy positive solutions of Schrödinger systems with mixed competition and cooperation terms, J. Differential Equations, 261 (2016), 505-537.  doi: 10.1016/j.jde.2016.03.015.  Google Scholar

[43]

M. Struwe, Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems. Fourth Edition, Springer-Verlag, Berlin, 2008.  Google Scholar

[44]

C. A. Stuart, Existence and approximation of solutions of nonlinear elliptic equations, Math. Z., 147 (1976), 53-63.  doi: 10.1007/BF01214274.  Google Scholar

[45]

S. Terracini, On positive entire solutions to a class of equations with a singular coefficient and critical exponent, Adv. Differential Equations, 1 (1996), 241-264.   Google Scholar

[46]

W. C. Troy, Symmetry properties in systems of semilinear elliptic equations, J. of Differential Equations, 42 (1981), 400-413.  doi: 10.1016/0022-0396(81)90113-3.  Google Scholar

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