Infinitely Many Sign-Changing Solutions for Some Nonlinear Fourth-Order Beam Equations

Abstract

Several new existence theorems on positive, negative, and sign-changing solutions for the following fourth-order beam equation are obtained: $u^{(4)}=f(t,u(t))$, $t\in [0,1]$; $u(0)=u(1)=u^{\prime \prime }(0)=u^{\prime \prime }(1)=0$, where $f\in C([0,1]\times {ℝ}^1,{ℝ}^1)$. In particular, an infinitely many sign changing solution theorem is established. The method of the invariant set of decreasing flow is employed to discuss this problem.