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Multiple solutions for a critical quasilinear equation with Hardy potential
Global symmetry-breaking bifurcations of critical orbits of invariant functionals
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 |
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.
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. |
[2] |
Z. Balanov, W. Krawcewicz and H. Steinlein, Applied Equivariant Degree, AIMS Series on Diff. Equat. & Dyn. Sys., Springfield, 2006. |
[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. |
[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. |
[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. |
[6] |
E. N. Dancer,
On nonradially symmetric bifurcation, J. London Math. Soc., 20 (1979), 287-292.
doi: 10.1112/jlms/s2-20.2.287. |
[7] |
E. N. Dancer,
Global breaking of symmetry of positive solutions on two-dimensional annuli, Diff. and Int. Equat., 5 (1992), 903-913.
|
[8] |
T. tom Dieck, Transformation Groups and Representation Theory, Lect. Not. in Math. 766 Springer, Berlin, 1979. |
[9] |
T. tom Dieck, Transformation Groups, Walter de Gruyter, Berlin, 1987.
doi: 10.1515/9783110858372.312. |
[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. |
[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. |
[12] |
K. Gȩba, M. Izydorek and A. Pruszko,
The Conley index in Hilbert spaces and its applications, Stud. Math., 134 (1999), 217-233.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[30] |
S. Rybicki,
Degree for equivariant gradient maps, Milan Journal of Mathematics, 73 (2005), 103-144.
doi: 10.1007/s00032-005-0040-2. |
[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. |
[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. |
[33] |
J. Smoller and A. Wasserman,
Bifurcation and symmetry-breaking, Invent. Math., 100 (1990), 63-95.
doi: 10.1007/BF01231181. |
[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. |
[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.
|
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. |
[2] |
Z. Balanov, W. Krawcewicz and H. Steinlein, Applied Equivariant Degree, AIMS Series on Diff. Equat. & Dyn. Sys., Springfield, 2006. |
[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. |
[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. |
[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. |
[6] |
E. N. Dancer,
On nonradially symmetric bifurcation, J. London Math. Soc., 20 (1979), 287-292.
doi: 10.1112/jlms/s2-20.2.287. |
[7] |
E. N. Dancer,
Global breaking of symmetry of positive solutions on two-dimensional annuli, Diff. and Int. Equat., 5 (1992), 903-913.
|
[8] |
T. tom Dieck, Transformation Groups and Representation Theory, Lect. Not. in Math. 766 Springer, Berlin, 1979. |
[9] |
T. tom Dieck, Transformation Groups, Walter de Gruyter, Berlin, 1987.
doi: 10.1515/9783110858372.312. |
[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. |
[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. |
[12] |
K. Gȩba, M. Izydorek and A. Pruszko,
The Conley index in Hilbert spaces and its applications, Stud. Math., 134 (1999), 217-233.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[30] |
S. Rybicki,
Degree for equivariant gradient maps, Milan Journal of Mathematics, 73 (2005), 103-144.
doi: 10.1007/s00032-005-0040-2. |
[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. |
[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. |
[33] |
J. Smoller and A. Wasserman,
Bifurcation and symmetry-breaking, Invent. Math., 100 (1990), 63-95.
doi: 10.1007/BF01231181. |
[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. |
[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.
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