TY - JOUR

T1 - Symmetry-breaking bifurcation for the one-dimensional Hénon equation

AU - Sim, Inbo

AU - Tanaka, Satoshi

N1 - Funding Information:
The first author work was supported by NRF Grant No. 2015R1D1A3A01019789 and second author work was supported by KAKENHI (26400182).
Publisher Copyright:
© 2019 World Scientific Publishing Company.

PY - 2019/2/1

Y1 - 2019/2/1

N2 - We show the existence of a symmetry-breaking bifurcation point for the one-dimensional Hénon equation u″ + |x|lup = 0, x ∈ (-1, 1), u(-1) = u(1) = 0, where l > 0 and p > 1. Moreover, employing a variant of Rabinowitz's global bifurcation, we obtain the unbounded connected set (the first of the alternatives about Rabinowitz's global bifurcation), which emanates from the symmetry-breaking bifurcation point. Finally, we give an example of a bounded branch connecting two symmetry-breaking bifurcation points (the second of the alternatives about Rabinowitz's global bifurcation) for the problem u″ + |x|l(λ)up = 0, x ∈ (-1, 1), u(-1) = u(1) = 0, where l is a specified continuous parametrization function.

AB - We show the existence of a symmetry-breaking bifurcation point for the one-dimensional Hénon equation u″ + |x|lup = 0, x ∈ (-1, 1), u(-1) = u(1) = 0, where l > 0 and p > 1. Moreover, employing a variant of Rabinowitz's global bifurcation, we obtain the unbounded connected set (the first of the alternatives about Rabinowitz's global bifurcation), which emanates from the symmetry-breaking bifurcation point. Finally, we give an example of a bounded branch connecting two symmetry-breaking bifurcation points (the second of the alternatives about Rabinowitz's global bifurcation) for the problem u″ + |x|l(λ)up = 0, x ∈ (-1, 1), u(-1) = u(1) = 0, where l is a specified continuous parametrization function.

KW - Hénon equation

KW - positive solution

KW - symmetry-breaking bifurcation

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U2 - 10.1142/S0219199717500973

DO - 10.1142/S0219199717500973

M3 - Article

AN - SCOPUS:85039167818

SN - 0219-1997

VL - 21

JO - Communications in Contemporary Mathematics

JF - Communications in Contemporary Mathematics

IS - 1

M1 - 1750097

ER -