Existence and Global Behavior of Positive Solutions for Some Fourth-Order Boundary Value Problems

Abstract

We establish the existence and uniqueness of a positive solution to the following fourth-order value problem: $u^{(4)}(x)=a(x)u^{\sigma }(x)$, $x\in (\mathrm{0,1})$ with the boundary conditions $u(0)=u(1)=u\mathrm{\text{'}}(0)=u\mathrm{\text{'}}(1)=0$, where $\sigma \in (-\mathrm{1,1})$ and $a$ is a nonnegative continuous function on (0, 1) that may be singular at $x=0$ or $x=1$. We also give the global behavior of such a solution.