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.


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