Asymptotic behavior of bifurcation curves of one-dimensional nonlocal elliptic equations

TL;DR

This paper investigates the asymptotic behavior of bifurcation curves in a one-dimensional nonlocal elliptic equation, providing global bifurcation diagram descriptions and precise asymptotic formulas for solutions as the bifurcation parameter grows large.

Contribution

It establishes the global structure of bifurcation diagrams and derives precise asymptotic formulas for solutions of a nonlocal elliptic equation as the bifurcation parameter tends to infinity.

Findings

Global bifurcation diagrams characterized

Asymptotic formulas for solutions derived

Behavior of solutions as λ → ∞ analyzed

Abstract

We study the one-dimensional nonlocal elliptic equation \begin{eqnarray*} -\left(\int_0^1 \vert u(x)\vert^p dx + b\right)^q u''(x) &=& \lambda u(x)^p, \quad x \in I:= (0,1), \ u(x) > 0, \ x\in I, \\ u(0) &=& u(1) = 0, \end{eqnarray*} where $b \ge 0, p \ge 1, q > 1 - \frac{1}{p}$ are given constants and $\lambda > 0$ is a bifurcation parameter. We establish the global behavior of bifurcation diagrams and precise asymptotic formulas for $u_\lambda(x)$ as $\lambda \to \infty$.