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Nodal solutions for a class of fourth-order two-point boundary value problems. (English) Zbl 1203.34034

Summary: We consider the fourth-order two-point boundary value problem \[ u'''+Mu=\lambda h(t)f(u),\quad 0<t<1, \]
\[ u(0)=u(1)=u'(0)=u'(1)=0, \]
where \(\lambda\in\mathbb R\) is a parameter, \(M\in (-\pi 4,\pi 4/64)\) is a given constant, \(h\in C([0,1],[0,\infty))\) with \(h(t)\not\equiv 0\) on any subinterval of \([0,1]\), \(f\in C(\mathbb R,\mathbb R)\) satisfies \(f(u)u>0\) for all \(u\neq 0\), and \(\lim_{u\to -\infty} f(u)/u=0\), \(\lim_{u\to +\infty} f(u)/u=f_{+\infty}\), \(\lim_{u\to 0}f(u)/u=f_0\) for some \(f_{+\infty}, f_0\in (0,+\infty)\). By using disconjugate operator theory and bifurcation techniques, we establish existence and multiplicity results of nodal solutions for the above problem.

MSC:

34B15 Nonlinear boundary value problems for ordinary differential equations
34C23 Bifurcation theory for ordinary differential equations
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References:

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