Double bifurcation diagrams and four positive solutions of nonlinear boundary value problems via time maps

Zhang, Xuemei; Feng, Meiqiang

doi:10.3934/cpaa.2018103

Cited by 4 publications

(3 citation statements)

References 22 publications

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“…Moreover, since q(ρ 1 , c 1 ) = 0 and lim ρ→(ρ1) + ∂ ∂ρ q(ρ, c 1 ) = f (ρ 1 ) by (26), (35) and (37), we have that…”

Section: Theorem 24 (Seementioning

confidence: 91%

Classification and evolution of bifurcation curves for a one-dimensional Dirichlet-Neumann problem with a specific cubic nonlinearity

Liang¹,

Wang²

2020

Discrete &Amp; Continuous Dynamical Systems - A

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We study the classification and evolution of bifurcation curves of positive solutions of the one-dimensional Dirichlet-Neumann problem with a specific cubic nonlinearity given by u (x) + λ(−εu 3 + u 2 + u + 1) = 0, 0 < x < 1, u(0) = 0, u (1) = −c < 0, where 1/10 ≤ ε ≤ 1/5. It is interesting to find that the evolution of bifurcation curves is not completely identical with that for the one-dimensional perturbed Gelfand equations, even though it is the same for these two problems with zero Dirichlet boundary conditions. In fact, we prove that there exist a positive number ε * (≈ 0.178) and three nonnegative numbers c 0 (ε) < c 1 (ε) < c 2 (ε) defined on [1/10, 1/5] with c 0 = 0 if 1/10 < ε ≤ ε * and c 0 > 0 if ε * < ε ≤ 1/5, such that, on the (λ, u ∞)-plane, (i) when 0 < c ≤ c 0 (ε) and c ≥ c 2 (ε), the bifurcation curve is strictly increasing; (ii) when c 0 (ε) < c < c 1 (ε), the bifurcation curve is S-shaped; (iii) when c 1 (ε) ≤ c < c 2 (ε), the bifurcation curve is ⊂-shaped.

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“…Moreover, since q(ρ 1 , c 1 ) = 0 and lim ρ→(ρ1) + ∂ ∂ρ q(ρ, c 1 ) = f (ρ 1 ) by (26), (35) and (37), we have that…”

Section: Theorem 24 (Seementioning

confidence: 91%

Classification and evolution of bifurcation curves for a one-dimensional Dirichlet-Neumann problem with a specific cubic nonlinearity

Liang¹,

Wang²

2020

Discrete &Amp; Continuous Dynamical Systems - A

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“…The study of boundary value problems with positive solutions has attracted recently the attention of different researchers and it is a topic of current interest; see [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16], and the references therein.…”

Section: Introductionmentioning

confidence: 99%

Positive radial solutions of n-dimensional elliptic systems with indefinite weight functions and n parameters

Feng

2019

Adv Differ Equ

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“…At the same time, we see that increasing attention has been paid to the study of nonlocal boundary value problems with parameters (see [21][22][23][24][25][26][27][28][29][30]). Especially, let us review several excellent results related to boundary value problems with parameters.…”

Section: Introductionmentioning

confidence: 99%

A class of nonlocal indefinite differential systems

Feng

Qin

2018

Bound Value Probl

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Double bifurcation diagrams and four positive solutions of nonlinear boundary value problems via time maps

Cited by 4 publications

References 22 publications

Classification and evolution of bifurcation curves for a one-dimensional Dirichlet-Neumann problem with a specific cubic nonlinearity

Classification and evolution of bifurcation curves for a one-dimensional Dirichlet-Neumann problem with a specific cubic nonlinearity

Positive radial solutions of n-dimensional elliptic systems with indefinite weight functions and n parameters

A class of nonlocal indefinite differential systems

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