TY - JOUR

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

AU - Kajikiya, Ryuji

AU - Sim, Inbo

AU - Tanaka, Satoshi

N1 - 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 -