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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 $[0, r]$, $f\in C(\bbfR, \bbfR)$ satisfies $f(u)u> 0$ for all $u\ne 0$, 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.

34B15Nonlinear boundary value problems for ODE
34C23Bifurcation (ODE)
Full Text: DOI
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