June  2017, 22(4): 1509-1523. doi: 10.3934/dcdsb.2017072

Morse indices and symmetry breaking for the Gelfand equation in expanding annuli

Department of Mathematics, Henan Normal University, Xinxiang 453007, China

* Corresponding author

Received  February 2016 Revised  November 2016 Published  February 2017

Fund Project: The research of the first author is supported by NSFC (11371117) and Startup Fund for Doctors of Henan Normal University (qd14154); the research of the second author is supported by NSFC (11171092, 11571093).

Bifurcation of nonradial solutions from radial solutions of
$-Δ u=λ e^u$
in expanding annuli of ${\mathbb{R}^N}$ with $3 ≤q N ≤q 9$ is studied. To obtain the main results, we use a blow-up argument via Morse indices of the regular entire solutions of (0.1).
Citation: Linfeng Mei, Zongming Guo. Morse indices and symmetry breaking for the Gelfand equation in expanding annuli. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1509-1523. doi: 10.3934/dcdsb.2017072
References:
[1]

T. BartschM. ClappM. Grossi and F. Pacella, Asymptotically radial solutions in expanding annular domains, Math. Ann., 352 (2012), 485-515.  doi: 10.1007/s00208-011-0646-3.  Google Scholar

[2]

M. G. Crandall and P. H. Rabinowitz, Some continuation and variational methods for positive solutions of nonlinear elliptic eigenvalue problems, Arch. Rational Mech. Anal., 58 (1975), 207-218.  doi: 10.1007/BF00280741.  Google Scholar

[3]

E. N. Dancer and A. Faria, On the classification of solutions of $-Δ u=e^u$ on ${\mathbb{R}^N}$ : stability outside a compact set and applications, Proc. Amer. Math. Soc., 137 (2009), 1333-1338.  doi: 10.1090/S0002-9939-08-09772-4.  Google Scholar

[4]

L. Dupaigne, Stable Solutions of Elliptic Partial Differential Equations Chapman & Hall / CRC Monographs and Surveys in Pure and Applied Mathematics, 143 2011. doi: 10.1201/b10802.  Google Scholar

[5]

A. Farina, Stable solutions of $-Δ u=e^u$ on ${\mathbb{R}^N}$, C. R. Acad. Sci. Paris, 345 (2007), 63-66.  doi: 10.1016/j.crma.2007.05.021.  Google Scholar

[6]

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

[7]

F. GladialiM. GrossiF. Pacella and P. N. Srikanth, Bifurcation and symmetry breaking for a class of semilinear elliptic equations in an annulus, Calc. Var. PDEs, 40 (2011), 295-317.  doi: 10.1007/s00526-010-0341-3.  Google Scholar

[8]

Y. Y. Li, Existence of many positive solutions of semilinear elliptic equations on annulus, J. Differential Equations, 83 (1990), 348-376.  doi: 10.1016/0022-0396(90)90062-T.  Google Scholar

[9]

S. S. Lin, Asymptotic behavior of positive solutions to semilinear elliptic equations on expanding annuli, J. Differential Equations, 120 (1995), 255-288.  doi: 10.1006/jdeq.1995.1112.  Google Scholar

[10]

S. S. Lin, Existence of many positive nonradial solutions for nonlinear elliptic equations on an annulus, J. Differential Equations, 103 (1993), 338-349.  doi: 10.1006/jdeq.1993.1053.  Google Scholar

[11]

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

[12]

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

[13]

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

[14]

K. Nagasaki and T. Suzuki, Radial solutions of $Δ u + λ e^u = 0$ on annuli in higher demensions, J. Differential Equations, 100 (1992), 137-161.  doi: 10.1016/0022-0396(92)90129-B.  Google Scholar

[15]

K. Nagasaki and T. Suzuki, Spectral and related properties about the Emdent-Fower equation $Δ u + λ e^u = 0$ on circular domains, Math. Ann., 299 (1994), 1-15.  doi: 10.1007/BF01459770.  Google Scholar

[16]

R. D. Nussbaum, The fixed point index for local condensing maps, Ann. Mat. Pura Appl., 89 (1971), 217-258.  doi: 10.1007/BF02414948.  Google Scholar

[17]

P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Funct. Anal., 7 (1971), 487-513.  doi: 10.1016/0022-1236(71)90030-9.  Google Scholar

[18]

P. H. Rabinowitz, Some aspects of nonlinear eigenvalue problems, Rocky Mountain J. Math., 3 (1973), 161-202.  doi: 10.1216/RMJ-1973-3-2-161.  Google Scholar

[19]

J. Smoller and A. Wasserman, Symmetry-breaking for solutions of semilinear elliptic equations with general boundary conditions, Comm. Math. Phys., 105 (1986), 415-441.  doi: 10.1007/BF01205935.  Google Scholar

show all references

References:
[1]

T. BartschM. ClappM. Grossi and F. Pacella, Asymptotically radial solutions in expanding annular domains, Math. Ann., 352 (2012), 485-515.  doi: 10.1007/s00208-011-0646-3.  Google Scholar

[2]

M. G. Crandall and P. H. Rabinowitz, Some continuation and variational methods for positive solutions of nonlinear elliptic eigenvalue problems, Arch. Rational Mech. Anal., 58 (1975), 207-218.  doi: 10.1007/BF00280741.  Google Scholar

[3]

E. N. Dancer and A. Faria, On the classification of solutions of $-Δ u=e^u$ on ${\mathbb{R}^N}$ : stability outside a compact set and applications, Proc. Amer. Math. Soc., 137 (2009), 1333-1338.  doi: 10.1090/S0002-9939-08-09772-4.  Google Scholar

[4]

L. Dupaigne, Stable Solutions of Elliptic Partial Differential Equations Chapman & Hall / CRC Monographs and Surveys in Pure and Applied Mathematics, 143 2011. doi: 10.1201/b10802.  Google Scholar

[5]

A. Farina, Stable solutions of $-Δ u=e^u$ on ${\mathbb{R}^N}$, C. R. Acad. Sci. Paris, 345 (2007), 63-66.  doi: 10.1016/j.crma.2007.05.021.  Google Scholar

[6]

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

[7]

F. GladialiM. GrossiF. Pacella and P. N. Srikanth, Bifurcation and symmetry breaking for a class of semilinear elliptic equations in an annulus, Calc. Var. PDEs, 40 (2011), 295-317.  doi: 10.1007/s00526-010-0341-3.  Google Scholar

[8]

Y. Y. Li, Existence of many positive solutions of semilinear elliptic equations on annulus, J. Differential Equations, 83 (1990), 348-376.  doi: 10.1016/0022-0396(90)90062-T.  Google Scholar

[9]

S. S. Lin, Asymptotic behavior of positive solutions to semilinear elliptic equations on expanding annuli, J. Differential Equations, 120 (1995), 255-288.  doi: 10.1006/jdeq.1995.1112.  Google Scholar

[10]

S. S. Lin, Existence of many positive nonradial solutions for nonlinear elliptic equations on an annulus, J. Differential Equations, 103 (1993), 338-349.  doi: 10.1006/jdeq.1993.1053.  Google Scholar

[11]

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

[12]

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

[13]

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

[14]

K. Nagasaki and T. Suzuki, Radial solutions of $Δ u + λ e^u = 0$ on annuli in higher demensions, J. Differential Equations, 100 (1992), 137-161.  doi: 10.1016/0022-0396(92)90129-B.  Google Scholar

[15]

K. Nagasaki and T. Suzuki, Spectral and related properties about the Emdent-Fower equation $Δ u + λ e^u = 0$ on circular domains, Math. Ann., 299 (1994), 1-15.  doi: 10.1007/BF01459770.  Google Scholar

[16]

R. D. Nussbaum, The fixed point index for local condensing maps, Ann. Mat. Pura Appl., 89 (1971), 217-258.  doi: 10.1007/BF02414948.  Google Scholar

[17]

P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Funct. Anal., 7 (1971), 487-513.  doi: 10.1016/0022-1236(71)90030-9.  Google Scholar

[18]

P. H. Rabinowitz, Some aspects of nonlinear eigenvalue problems, Rocky Mountain J. Math., 3 (1973), 161-202.  doi: 10.1216/RMJ-1973-3-2-161.  Google Scholar

[19]

J. Smoller and A. Wasserman, Symmetry-breaking for solutions of semilinear elliptic equations with general boundary conditions, Comm. Math. Phys., 105 (1986), 415-441.  doi: 10.1007/BF01205935.  Google Scholar

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