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November  2019, 12(7): 2005-2017. doi: 10.3934/dcdss.2019129

Global symmetry-breaking bifurcations of critical orbits of invariant functionals

Anna Goƚȩbiewska 1,, , Norimichi Hirano 2, and  Sƚawomir Rybicki 1,

1. 

Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, PL-87-100 Toruń, ul. Chopina 12/18, Poland
2. 

Department of Mathematics, Graduate School of Environment and Information Sciences, Yokohama National University, 79-7 Tokiwadai, Hodogayaku, Yokohama, Japan

* Corresponding author

Received  October 2017 Revised  April 2018 Published  December 2018

Fund Project: Partially supported by the National Science Center, Poland, under grant DEC-2012/05/B/ST1/02165.

In this article we present a method of study of a global symmetry-breaking bifurcation of critical orbits of invariant functionals. As a topological tool we use the degree for equivariant gradient maps. We underline that many known results on bifurcations of non-radial solutions of elliptic PDE's from the families of radial ones are consequences of our theory.

Keywords: Equivariant bifurcation theory, global bifurcation, symmetry-breaking bifurcation, degree for equivariant gradient maps, non-radial bifurcations.
Mathematics Subject Classification: Primary: 58E09; Secondary: 35J50.
Citation: Anna Goƚȩbiewska, Norimichi Hirano, Sƚawomir Rybicki. Global symmetry-breaking bifurcations of critical orbits of invariant functionals. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 2005-2017. doi: 10.3934/dcdss.2019129
References:
[1]

Z. Balanov, W. Krawcewicz, S. Rybicki and H. Steinlein, A short treatise on the equivariant degree theory and its applications, J. Fixed P. Th. and Appl., 8 (2010), 1-74.  doi: 10.1007/s11784-010-0033-9.  Google Scholar

[2]

Z. Balanov, W. Krawcewicz and H. Steinlein, Applied Equivariant Degree, AIMS Series on Diff. Equat. & Dyn. Sys., Springfield, 2006.  Google Scholar

[3]

P. Bartlomiejczyk, K. Gȩba and M. Izydorek, Otopy classes of equivariant maps, J. Fixed P. Th. and Appl., 7 (2010), 145-160.  doi: 10.1007/s11784-010-0013-0.  Google Scholar

[4]

G. Cerami, Symmetry breaking for a class of semilinear elliptic problems, Nonl. Anal. TMA, 10 (1986), 1-14.  doi: 10.1016/0362-546X(86)90007-6.  Google Scholar

[5]

M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Funct. Anal., 8 (1971), 321-340.  doi: 10.1016/0022-1236(71)90015-2.  Google Scholar

[6]

E. N. Dancer, On nonradially symmetric bifurcation, J. London Math. Soc., 20 (1979), 287-292.  doi: 10.1112/jlms/s2-20.2.287.  Google Scholar

[7]

E. N. Dancer, Global breaking of symmetry of positive solutions on two-dimensional annuli, Diff. and Int. Equat., 5 (1992), 903-913.   Google Scholar

[8]

T. tom Dieck, Transformation Groups and Representation Theory, Lect. Not. in Math. 766 Springer, Berlin, 1979.  Google Scholar

[9]

T. tom Dieck, Transformation Groups, Walter de Gruyter, Berlin, 1987. doi: 10.1515/9783110858372.312.  Google Scholar

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J. Fura, A. Ratajczak and S. Rybicki, Existence and continuation of periodic solutions of autonomous Newtonian systems, J. Diff. Equat., 218 (2005), 216-252.  doi: 10.1016/j.jde.2005.04.004.  Google Scholar

[11]

K. Gȩba, Degree for gradient equivariant maps and equivariant Conley index, Topological Nonlinear Analysis, Degree, Singularity and Variations, Eds. M. Matzeu & A. Vignoli, Progr. in Nonl. Diff. Equat. and Their Appl., 27, Birkhäuser, (1997), 247-272.  Google Scholar

[12]

K. Gȩba, M. Izydorek and A. Pruszko, The Conley index in Hilbert spaces and its applications, Stud. Math., 134 (1999), 217-233.   Google Scholar

[13]

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.  Google Scholar

[14]

A. Golȩbiewska and S. Rybicki, Global bifurcations of critical orbits of $G$-invariant strongly indefinite functionals, Nonl. Anal. TMA, 74 (2011), 1823-1834.  doi: 10.1016/j.na.2010.10.055.  Google Scholar

[15]

W. Jäger and K. Schmitt, Symmetry breaking in semilinear elliptic problems, Academic Press, Inc., Analysis, et cetera, Research Papers Published in Honour of Jürgen Moser's 60th Birthday, Eds. P. H. Rabinowitz & E. Zehnder, (1990), 451-470. Google Scholar

[16]

R. Lauterbach and S. Maier, Symmetry-breaking at non-positive solutions of semilinear elliptic equations, Arch. Rational Mech. Anal., 126 (1994), 299-331.  doi: 10.1007/BF00380895.  Google Scholar

[17]

Y. Y. Li, Existence of many positive solutions of semilinear elliptic equations on annulus, J. Diff. Equat., 83 (1990), 348-367.  doi: 10.1016/0022-0396(90)90062-T.  Google Scholar

[18]

S. S. Lin, On non-radially symmetric bifurcation in the annulus, J. Diff. Equat., 80 (1989), 251-279.  doi: 10.1016/0022-0396(89)90084-3.  Google Scholar

[19]

S. S. Lin, Positive radial solutions and non-radial bifurcation for semilinear elliptic equations in annular domains, J. Diff. Equat., 86 (1990), 367-391.  doi: 10.1016/0022-0396(90)90035-N.  Google Scholar

[20]

S. S. Lin, Existence of positive non-radial solutions for nonlinear elliptic equations in annular domains, Trans. of the Amer. Math. Soc., 332 (1992), 775-791.  doi: 10.1090/S0002-9947-1992-1055571-1.  Google Scholar

[21]

G. López Garza and S. Rybicki, Equivariant bifurcation index, Nonl. Anal. TMA, 73 (2010), 2779-2791.  doi: 10.1016/j.na.2010.06.001.  Google Scholar

[22]

K. Nagasaki and T. Suzuki, Radial and nonradial solutions for the nonlinear eigenvalue problem $\Delta u + \lambda e^u = 0$ on annuli in $R^2$, J. Diff. Equat., 87 (1990), 144-168.  doi: 10.1016/0022-0396(90)90020-P.  Google Scholar

[23]

K. Nagasaki and T. Suzuki, Radial Solutions for $\Delta u + \lambda e^u = 0$ on annuli in higher dimensions, J. Diff. Equat., 100 (1992), 137-161.  doi: 10.1016/0022-0396(92)90129-B.  Google Scholar

[24]

K. Nagasaki and T. Suzuki, Spectral and related properties about the Emden-Fowler equation $ \Delta u = \lambda e^u$ on circular domains, Math. Ann., 299 (1994), 1-15.  doi: 10.1007/BF01459770.  Google Scholar

[25]

W. -M. Ni and R. Nussbaum, Uniqueness and nonuniqueness for positive radial solutions of $-\Delta u+f(u, r)=0$, Comm. Pure Appl. Math., 38 (1985), 67-108.  doi: 10.1002/cpa.3160380105.  Google Scholar

[26]

F. Pacard, Radial and non-radial solutions of $-Δ u = λ f(u)$, on an annulus of $\mathbb{R}^n, \:n ≥ 3$, J. Diff. Equat., 101 (1993), 103-138.  doi: 10.1006/jdeq.1993.1007.  Google Scholar

[27]

M. Ramaswamy and P. N. Srikanth, Symmetry breaking for a class of semilinear elliptic problems, Trans. of the Amer. Math. Soc., 304 (1987), 839-845.  doi: 10.1090/S0002-9947-1987-0911098-4.  Google Scholar

[28]

S. Rybicki, A degree for $S^1$-equivariant orthogonal maps and its applications to bifurcation theory, Nonl. Anal. TMA, 23 (1994), 83-102.  doi: 10.1016/0362-546X(94)90253-4.  Google Scholar

[29]

S. Rybicki, Global bifurcations of solutions of Emden-Fowler type equation $- \Delta u(x) = \lambda f(u(x))$ on an annulus in $R^n, \: n \geq 3$, J. Diff. Equat., 183 (2002), 208-223.  doi: 10.1006/jdeq.2001.4121.  Google Scholar

[30]

S. Rybicki, Degree for equivariant gradient maps, Milan Journal of Mathematics, 73 (2005), 103-144.  doi: 10.1007/s00032-005-0040-2.  Google Scholar

[31]

S. Rybicki, Global bifurcations of critical orbits via equivariant Conley index, Adv. Nonl. Studies, 11 (2011), 929-940.  doi: 10.1515/ans-2011-0410.  Google Scholar

[32]

J. Smoller and A. G. Wasserman, Symmetry-breaking for positive solutions of semilinear elliptic equations, Arch. Rational Mech. Anal., 95 (1986), 217-225.  doi: 10.1007/BF00251359.  Google Scholar

[33]

J. Smoller and A. Wasserman, Bifurcation and symmetry-breaking, Invent. Math., 100 (1990), 63-95.  doi: 10.1007/BF01231181.  Google Scholar

[34]

P. N. Srikanth, Symmetry breaking for a class of semilinear elliptic problems, Ann. Inst. H. Poincarè Anal. Non Lin., 7 (1990), 107-112.  doi: 10.1016/S0294-1449(16)30301-8.  Google Scholar

[35]

T. Suzuki, Two-dimensional Emden-Fowler equation with exponential nonlinearity, Nonlinear diffusion equations and their equilibrium states, Proc. of 3rd Conf. Gregynog UK, 1989, Prog. Nonl. Diff. Equat. Appl., 7 (1992), 493-512.   Google Scholar

show all references

References:
[1]

Z. Balanov, W. Krawcewicz, S. Rybicki and H. Steinlein, A short treatise on the equivariant degree theory and its applications, J. Fixed P. Th. and Appl., 8 (2010), 1-74.  doi: 10.1007/s11784-010-0033-9.  Google Scholar

[2]

Z. Balanov, W. Krawcewicz and H. Steinlein, Applied Equivariant Degree, AIMS Series on Diff. Equat. & Dyn. Sys., Springfield, 2006.  Google Scholar

[3]

P. Bartlomiejczyk, K. Gȩba and M. Izydorek, Otopy classes of equivariant maps, J. Fixed P. Th. and Appl., 7 (2010), 145-160.  doi: 10.1007/s11784-010-0013-0.  Google Scholar

[4]

G. Cerami, Symmetry breaking for a class of semilinear elliptic problems, Nonl. Anal. TMA, 10 (1986), 1-14.  doi: 10.1016/0362-546X(86)90007-6.  Google Scholar

[5]

M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Funct. Anal., 8 (1971), 321-340.  doi: 10.1016/0022-1236(71)90015-2.  Google Scholar

[6]

E. N. Dancer, On nonradially symmetric bifurcation, J. London Math. Soc., 20 (1979), 287-292.  doi: 10.1112/jlms/s2-20.2.287.  Google Scholar

[7]

E. N. Dancer, Global breaking of symmetry of positive solutions on two-dimensional annuli, Diff. and Int. Equat., 5 (1992), 903-913.   Google Scholar

[8]

T. tom Dieck, Transformation Groups and Representation Theory, Lect. Not. in Math. 766 Springer, Berlin, 1979.  Google Scholar

[9]

T. tom Dieck, Transformation Groups, Walter de Gruyter, Berlin, 1987. doi: 10.1515/9783110858372.312.  Google Scholar

[10]

J. Fura, A. Ratajczak and S. Rybicki, Existence and continuation of periodic solutions of autonomous Newtonian systems, J. Diff. Equat., 218 (2005), 216-252.  doi: 10.1016/j.jde.2005.04.004.  Google Scholar

[11]

K. Gȩba, Degree for gradient equivariant maps and equivariant Conley index, Topological Nonlinear Analysis, Degree, Singularity and Variations, Eds. M. Matzeu & A. Vignoli, Progr. in Nonl. Diff. Equat. and Their Appl., 27, Birkhäuser, (1997), 247-272.  Google Scholar

[12]

K. Gȩba, M. Izydorek and A. Pruszko, The Conley index in Hilbert spaces and its applications, Stud. Math., 134 (1999), 217-233.   Google Scholar

[13]

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.  Google Scholar

[14]

A. Golȩbiewska and S. Rybicki, Global bifurcations of critical orbits of $G$-invariant strongly indefinite functionals, Nonl. Anal. TMA, 74 (2011), 1823-1834.  doi: 10.1016/j.na.2010.10.055.  Google Scholar

[15]

W. Jäger and K. Schmitt, Symmetry breaking in semilinear elliptic problems, Academic Press, Inc., Analysis, et cetera, Research Papers Published in Honour of Jürgen Moser's 60th Birthday, Eds. P. H. Rabinowitz & E. Zehnder, (1990), 451-470. Google Scholar

[16]

R. Lauterbach and S. Maier, Symmetry-breaking at non-positive solutions of semilinear elliptic equations, Arch. Rational Mech. Anal., 126 (1994), 299-331.  doi: 10.1007/BF00380895.  Google Scholar

[17]

Y. Y. Li, Existence of many positive solutions of semilinear elliptic equations on annulus, J. Diff. Equat., 83 (1990), 348-367.  doi: 10.1016/0022-0396(90)90062-T.  Google Scholar

[18]

S. S. Lin, On non-radially symmetric bifurcation in the annulus, J. Diff. Equat., 80 (1989), 251-279.  doi: 10.1016/0022-0396(89)90084-3.  Google Scholar

[19]

S. S. Lin, Positive radial solutions and non-radial bifurcation for semilinear elliptic equations in annular domains, J. Diff. Equat., 86 (1990), 367-391.  doi: 10.1016/0022-0396(90)90035-N.  Google Scholar

[20]

S. S. Lin, Existence of positive non-radial solutions for nonlinear elliptic equations in annular domains, Trans. of the Amer. Math. Soc., 332 (1992), 775-791.  doi: 10.1090/S0002-9947-1992-1055571-1.  Google Scholar

[21]

G. López Garza and S. Rybicki, Equivariant bifurcation index, Nonl. Anal. TMA, 73 (2010), 2779-2791.  doi: 10.1016/j.na.2010.06.001.  Google Scholar

[22]

K. Nagasaki and T. Suzuki, Radial and nonradial solutions for the nonlinear eigenvalue problem $\Delta u + \lambda e^u = 0$ on annuli in $R^2$, J. Diff. Equat., 87 (1990), 144-168.  doi: 10.1016/0022-0396(90)90020-P.  Google Scholar

[23]

K. Nagasaki and T. Suzuki, Radial Solutions for $\Delta u + \lambda e^u = 0$ on annuli in higher dimensions, J. Diff. Equat., 100 (1992), 137-161.  doi: 10.1016/0022-0396(92)90129-B.  Google Scholar

[24]

K. Nagasaki and T. Suzuki, Spectral and related properties about the Emden-Fowler equation $ \Delta u = \lambda e^u$ on circular domains, Math. Ann., 299 (1994), 1-15.  doi: 10.1007/BF01459770.  Google Scholar

[25]

W. -M. Ni and R. Nussbaum, Uniqueness and nonuniqueness for positive radial solutions of $-\Delta u+f(u, r)=0$, Comm. Pure Appl. Math., 38 (1985), 67-108.  doi: 10.1002/cpa.3160380105.  Google Scholar

[26]

F. Pacard, Radial and non-radial solutions of $-Δ u = λ f(u)$, on an annulus of $\mathbb{R}^n, \:n ≥ 3$, J. Diff. Equat., 101 (1993), 103-138.  doi: 10.1006/jdeq.1993.1007.  Google Scholar

[27]

M. Ramaswamy and P. N. Srikanth, Symmetry breaking for a class of semilinear elliptic problems, Trans. of the Amer. Math. Soc., 304 (1987), 839-845.  doi: 10.1090/S0002-9947-1987-0911098-4.  Google Scholar

[28]

S. Rybicki, A degree for $S^1$-equivariant orthogonal maps and its applications to bifurcation theory, Nonl. Anal. TMA, 23 (1994), 83-102.  doi: 10.1016/0362-546X(94)90253-4.  Google Scholar

[29]

S. Rybicki, Global bifurcations of solutions of Emden-Fowler type equation $- \Delta u(x) = \lambda f(u(x))$ on an annulus in $R^n, \: n \geq 3$, J. Diff. Equat., 183 (2002), 208-223.  doi: 10.1006/jdeq.2001.4121.  Google Scholar

[30]

S. Rybicki, Degree for equivariant gradient maps, Milan Journal of Mathematics, 73 (2005), 103-144.  doi: 10.1007/s00032-005-0040-2.  Google Scholar

[31]

S. Rybicki, Global bifurcations of critical orbits via equivariant Conley index, Adv. Nonl. Studies, 11 (2011), 929-940.  doi: 10.1515/ans-2011-0410.  Google Scholar

[32]

J. Smoller and A. G. Wasserman, Symmetry-breaking for positive solutions of semilinear elliptic equations, Arch. Rational Mech. Anal., 95 (1986), 217-225.  doi: 10.1007/BF00251359.  Google Scholar

[33]

J. Smoller and A. Wasserman, Bifurcation and symmetry-breaking, Invent. Math., 100 (1990), 63-95.  doi: 10.1007/BF01231181.  Google Scholar

[34]

P. N. Srikanth, Symmetry breaking for a class of semilinear elliptic problems, Ann. Inst. H. Poincarè Anal. Non Lin., 7 (1990), 107-112.  doi: 10.1016/S0294-1449(16)30301-8.  Google Scholar

[35]

T. Suzuki, Two-dimensional Emden-Fowler equation with exponential nonlinearity, Nonlinear diffusion equations and their equilibrium states, Proc. of 3rd Conf. Gregynog UK, 1989, Prog. Nonl. Diff. Equat. Appl., 7 (1992), 493-512.   Google Scholar

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