Constant sign solution for a simply supported beam equation

Abstract

The aim of this paper is to ensure the existence of constant sign solutions for the fourth order boundary value problem: \[\begin{cases} u^{(4)}(t)-p\,u''(t)+c(t)\,u(t)=h(t)(\geq 0)\,,&t\in I\equiv[a,b]\,,\\ u(a)=u''(a)=u(b)=u''(b)=0\,, \end{cases}\] where $c,\ h\in C(I)$ and $p\geq 0$. This problem models the behavior of a suspension bridge assuming that the vertical displacement is small enough. By using variational methods, we weaken the previously known sufficient conditions on $c$ to ensure that the obtained solution is of constant sign.

Keywords

• differential equations
• boundary value problems
• constant sign solutions
• variational approach