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Nodal solutions for a fourth-order two-point boundary value problem. (English) Zbl 1085.34015

Summary: We consider boundary value problems of fourth-order differential equations of the form \[ u''''+\beta u''-\alpha u=\mu h(x) f(u),\qquad 0< x< r, \]
\[ u(0)= u(r)= u''(0)= u''(r)= 0, \] where \(\mu\) is a parameter, \(\beta\in(-\infty, \infty)\), \(\alpha\in [0,\infty)\) are constants with \[ {r^2\beta\over\pi^2}+ {r^4\alpha\over\pi^4}< 1, \] \(h\in C(0, r], [0,\infty))\) with \(h\not\equiv 0\) on any subinterval of \([0, r]\), \(f\in C(\mathbb{R}, \mathbb{R})\) satisfies \(f(u)u> 0\) for all \(u\neq 0\), and \[ \lim_{u\to-\infty} {f(u)\over u}= 0,\quad \lim_{u\to+\infty} {f(u)\over u}= f_{+\infty},\quad \lim_{u\to 0} {f(u)\over u}= f_0, \] for some \(f_{+\infty}\), \(f_0\in (0,\infty)\). We use bifurcation techniques to establish existence and multiplicity results on nodal solutions to the 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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