TY - JOUR

T1 - Symmetry-breaking bifurcation for the Moore–Nehari differential equation

AU - Kajikiya, Ryuji

AU - Sim, Inbo

AU - Tanaka, Satoshi

N1 - Funding Information:
Ryuji Kajikiya was supported by JSPS KAKENHI Grant Number 16K05236. Inbo Sim was supported by NRF Grant No. 2015R1D1A3A01019789. Satoshi Tanaka was supported by JSPS KAKENHI Grant Number 26400182.
Publisher Copyright:
© 2018, Springer Nature Switzerland AG.

PY - 2018/12/1

Y1 - 2018/12/1

N2 - We study the bifurcation problem of positive solutions for the Moore-Nehari differential equation, u′ ′+ h(x, λ) up= 0 , u> 0 in (- 1 , 1) with u(- 1) = u(1) = 0 , where p> 1 , h(x, λ) = 0 for | x| < λ and h(x, λ) = 1 for λ≤ | x| ≤ 1 and λ∈ (0 , 1) is a bifurcation parameter. We shall show that the problem has a unique even positive solution U(x, λ) for each λ∈ (0 , 1). We shall prove that there exists a unique λ∗∈ (0 , 1) such that a non-even positive solution bifurcates at λ∗ from the curve (λ, U(x, λ)) , where λ∗ is explicitly represented as a function of p.

AB - We study the bifurcation problem of positive solutions for the Moore-Nehari differential equation, u′ ′+ h(x, λ) up= 0 , u> 0 in (- 1 , 1) with u(- 1) = u(1) = 0 , where p> 1 , h(x, λ) = 0 for | x| < λ and h(x, λ) = 1 for λ≤ | x| ≤ 1 and λ∈ (0 , 1) is a bifurcation parameter. We shall show that the problem has a unique even positive solution U(x, λ) for each λ∈ (0 , 1). We shall prove that there exists a unique λ∗∈ (0 , 1) such that a non-even positive solution bifurcates at λ∗ from the curve (λ, U(x, λ)) , where λ∗ is explicitly represented as a function of p.

KW - Bifurcation

KW - Morse index

KW - Positive solution

KW - Symmetry-breaking

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U2 - 10.1007/s00030-018-0545-3

DO - 10.1007/s00030-018-0545-3

M3 - Article

AN - SCOPUS:85057327575

SN - 1021-9722

VL - 25

JO - Nonlinear Differential Equations and Applications

JF - Nonlinear Differential Equations and Applications

IS - 6

M1 - 54

ER -