Nondegeneracy of the second bifurcating branches for the ChafeeInfante problem on a planar symmetric domain
Author:
Yasuhito Miyamoto
Journal:
Proc. Amer. Math. Soc. 139 (2011), 975984
MSC (2010):
Primary 35B32, 35P15; Secondary 35J61, 35J15
Published electronically:
July 30, 2010
MathSciNet review:
2745649
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Additional Information
Abstract: Let be a planar domain such that is symmetric with respect to both the  and axes and satisfies certain conditions. Then the second eigenvalue of the Dirichlet Laplacian on , , is simple, and the corresponding eigenfunction is odd with respect to the axis. Let be a function such that Let denote the maximal continua consisting of nontrivial solutions, , to and emanating from the second eigenvalue . We show that, for each , the Morse index of is one and zero is not an eigenvalue of the linearized problem. We show that consists of two unbounded curves, each curve is parametrized by and the closure is homeomorphic to .
 1.
T. Carleman, Sur les systèmes linéaires aux derivées partielles du premier ordre à deux variables, C. R. Acad. Sci. Paris 197 (1933), 471474.
 2.
N.
Chafee and E.
F. Infante, A bifurcation problem for a nonlinear partial
differential equation of parabolic type, Applicable Anal.
4 (1974/75), 17–37. MR 0440205
(55 #13084)
 3.
Michael
G. Crandall and Paul
H. Rabinowitz, Bifurcation from simple eigenvalues, J.
Functional Analysis 8 (1971), 321–340. MR 0288640
(44 #5836)
 4.
Manuel
del Pino, Jorge
GarcíaMelián, and Monica
Musso, Local bifurcation from the second
eigenvalue of the Laplacian in a square, Proc.
Amer. Math. Soc. 131 (2003), no. 11, 3499–3505 (electronic). MR 1991761
(2004c:35027), 10.1090/S0002993903069065
 5.
B.
Gidas, Wei
Ming Ni, and L.
Nirenberg, Symmetry and related properties via the maximum
principle, Comm. Math. Phys. 68 (1979), no. 3,
209–243. MR
544879 (80h:35043)
 6.
Marco
Holzmann and Hansjörg
Kielhöfer, Uniqueness of global positive solution branches of
nonlinear elliptic problems, Math. Ann. 300 (1994),
no. 2, 221–241. MR 1299061
(95m:35068), 10.1007/BF01450485
 7.
Philip
Hartman and Aurel
Wintner, On the local behavior of solutions of nonparabolic
partial differential equations, Amer. J. Math. 75
(1953), 449–476. MR 0058082
(15,318b)
 8.
Philip
Korman, Solution curves for semilinear
equations on a ball, Proc. Amer. Math. Soc.
125 (1997), no. 7,
1997–2005. MR 1423311
(97j:35037), 10.1090/S0002993997041191
 9.
Yasuhito
Miyamoto, Global branches of nonradially symmetric solutions to a
semilinear Neumann problem in a disk, J. Funct. Anal.
256 (2009), no. 3, 747–776. MR 2484935
(2010d:35130), 10.1016/j.jfa.2008.11.023
 10.
Yasuhito
Miyamoto, Nonexistence of a secondary bifurcation point for a
semilinear elliptic problem in the presence of symmetry, J. Math.
Anal. Appl. 357 (2009), no. 1, 89–97. MR 2526808
(2010d:35131), 10.1016/j.jmaa.2009.04.005
 11.
Y. Miyamoto, Global branches of signchanging solutions to a semilinear Dirichlet problem in a disk, preprint.
 12.
Y. Mukai, Bifurcation of solutions to a boundary value problem of a semilinear elliptic equation on a disk (in Japanese), Master Thesis (1997), Graduate School of Mathematical Sciences, University of Tokyo, Japan.
 13.
Rolf
Pütter, On the nodal lines of second eigenfunctions of the
fixed membrane problem, Comment. Math. Helv. 65
(1990), no. 1, 96–103. MR 1036131
(91c:35109), 10.1007/BF02566596
 14.
Chao
Liang Shen, Remarks on the second eigenvalue of a symmetric simply
connected plane region, SIAM J. Math. Anal. 19
(1988), no. 1, 167–171. MR 924553
(89a:35152), 10.1137/0519013
 15.
José
M. Vegas, Bifurcations caused by perturbing the domain in an
elliptic equation, J. Differential Equations 48
(1983), no. 2, 189–226. MR 696867
(84g:35020), 10.1016/00220396(83)900499
 1.
 T. Carleman, Sur les systèmes linéaires aux derivées partielles du premier ordre à deux variables, C. R. Acad. Sci. Paris 197 (1933), 471474.
 2.
 N. Chafee and E. Infante, A bifurcation problem for a nonlinear partial differential equation of parabolic type, Applicable Anal. 4 (1974/75), 1737. MR 0440205 (55:13084)
 3.
 M. Crandall and P. Rabinowitz, Bifurcation from simple eigenvalues, J. Funct. Anal. 8 (1971), 321340. MR 0288640 (44:5836)
 4.
 M. del Pino, J. GarcíaMelián and M. Musso, Local bifurcation from the second eigenvalue of the Laplacian in a square, Proc. Amer. Math. Soc. 131 (2003), 34993505. MR 1991761 (2004c:35027)
 5.
 B. Gidas, W.M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68 (1979), 209243. MR 544879 (80h:35043)
 6.
 M. Holzmann and H. Kielhöfer, Uniqueness of global positive solution branches of nonlinear elliptic problems, Math. Ann. 300 (1994), 221241. MR 1299061 (95m:35068)
 7.
 P. Hartman and A. Wintner, On the local behavior of solutions of nonparabolic partial differential equations, Amer. J. Math. 75 (1953), 449476. MR 0058082 (15:318b)
 8.
 P. Korman, Solution curves for semilinear equations on a ball, Proc. Amer. Math. Soc. 125 (1997), 19972005. MR 1423311 (97j:35037)
 9.
 Y. Miyamoto, Global branches of nonradially symmetric solutions to a semilinear Neumann problem in a disk, J. Funct. Anal. 256 (2009), 747776. MR 2484935 (2010d:35130)
 10.
 Y. Miyamoto, Nonexistence of a secondary bifurcation point for a semilinear elliptic problem in the presence of symmetry, J. Math. Anal. Appl. 357 (2009), 8997. MR 2526808 (2010d:35131)
 11.
 Y. Miyamoto, Global branches of signchanging solutions to a semilinear Dirichlet problem in a disk, preprint.
 12.
 Y. Mukai, Bifurcation of solutions to a boundary value problem of a semilinear elliptic equation on a disk (in Japanese), Master Thesis (1997), Graduate School of Mathematical Sciences, University of Tokyo, Japan.
 13.
 R. Pütter, On the nodal lines of second eigenfunctions of the fixed membrane problem, Comment. Math. Helv. 65 (1990), 96103. MR 1036131 (91c:35109)
 14.
 C. Shen, Remarks on the second eigenvalue of a symmetric simply connected plane region, SIAM J. Math. Anal. 19 (1988), 167171. MR 924553 (89a:35152)
 15.
 J. Vegas, Bifurcations caused by perturbing the domain in an elliptic equation, J. Differential Equations 48 (1983), 189226. MR 696867 (84g:35020)
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Additional Information
Yasuhito Miyamoto
Affiliation:
Department of Mathematics, Tokyo Institute of Technology, Meguroku, Tokyo 1528551, Japan
Email:
miyamoto@math.titech.ac.jp
DOI:
http://dx.doi.org/10.1090/S000299392010106160
Keywords:
Global branch,
bifurcation,
signchanging solution,
planar symmetric domain,
second eigenvalue
Received by editor(s):
March 23, 2010
Published electronically:
July 30, 2010
Additional Notes:
This work was partially supported by GrantinAid for Young Scientists (B) (Subject No. 21740116).
Communicated by:
Matthew J. Gursky
Article copyright:
© Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
