Nodal solutions of boundary value problems of fourth-order ordinary differential equations. (English) Zbl 1098.34012

Summary: We study the existence of nodal solutions of the fourth-order two-point boundary value problem \[ y''''+\beta(t)y''=a(t)f(y),\;0<t<1,\qquad y(0)=y(1)=y''(1)=0, \] where \(\beta\in C[0,1]\) with \(\beta(t)<\pi^2\) on \([0,1]\), \(a\in C[0,1]\) with \(a\geq 0\) on \([0,1]\) and \(a(t)\equiv 0\) on any subinterval of \([0,1]\), and \(f\in C(\mathbb{R})\) satisfies \(f(u)u>0\) for all \(u\neq 0\). We give conditions on the ratio \(f(s)/s\) at infinity and zero that guarantee the existence of nodal solutions. The proof of our main results is based upon bifurcation techniques.


34B15 Nonlinear boundary value problems for ordinary differential equations
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