S-shaped bifurcation curve for a nonlocal boundary value problem

Xu, Xianmin; Qin, Baoxia; Li, Wei

doi:10.1016/j.jmaa.2016.12.073

Cited by 3 publications

(1 citation statement)

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“…For other results concerning the existence of an S-shaped connected component in the set of solutions for diverse boundary value problems, see [23][24][25][26] for the semilinear boundary value problems, and [27] for the p-Laplacian boundary value problems.…”

Section: Introductionmentioning

confidence: 99%

S-shaped connected component of radial positive solutions for a prescribed mean curvature problem in an annular domain

2019

Open Mathematics

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In this paper, we show the existence of an S-shaped connected component in the set of radial positive solutions of boundary value problem$$\begin{array}{} \displaystyle \left\{\,\begin{array}{} -\text{ div}\big(\phi_N(\nabla y)\big)=\lambda a(|x|)f(y)\, \, \, \, \, \text{in}\, \, \mathcal{A},\\\frac{\partial y}{\partial \nu}=0\, \, \, \,\, \text{ on }\, \, {\it\Gamma}_1,\qquad y=0\, \, \, \, \text{ on}\, \, {\it\Gamma}_2,\\ \end{array} \right. \end{array} $$where R2 ∈ (0, ∞) and R1 ∈ (0, R2) is a given constant, 𝓐 = {x ∈ ℝN : R1 < ∣x∣ < R2}, Γ1 = {x ∈ ℝN : ∣x∣ = R1}, Γ2 = {x ∈ ℝN : ∣x∣ = R2}, $\begin{array}{} \phi_N(s)=\frac{s}{\sqrt{1-|s|^2}}, \end{array} $s ∈ ℝN, λ is a positive parameter, a ∈ C[R1, R2], f ∈ C[0, ∞), $\begin{array}{} \frac{\partial y}{\partial \nu} \end{array} $ denotes the outward normal derivative of y and ∣⋅∣ denotes the Euclidean norm in ℝN. The proof of main result is based upon bifurcation techniques.

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