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

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 $\lambda \to \infty$.