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Zbl 1017.34015
Rynne, Bryan P.
Infinitely many solutions of superlinear fourth order boundary value problems.
(English)
[J] Topol. Methods Nonlinear Anal. 19, No.2, 303-312 (2002). ISSN 1230-3429

The author considers the boundary value problem $$u^{(4)}(x)= g(u(x))+ p(x, u^{(0)}(x),\dots, u^{(3)}(x)),\quad x\in (0,1),$$ $$u(0)= u(1)= u^{(b)}(0)= u^{(b)}(1)= 0,$$ where\par (i) $g:\bbfR\to \bbfR$ is continuous and satisfies $\lim_{|\xi|\to\infty} g(\xi)/\xi= \infty$ ($g$ is super-linear as $|\xi|\to\infty$),\par (ii) $p: [0,1]\times \bbfR^4\to\bbfR$ is continuous and satisfies $|p(x,\xi_0,\xi_1,\xi_2,\xi_3)|\le C+{1\over 4}|\xi_0|$, $x\in [0,1]$, $(\xi_0,\xi_1,\xi_2,\xi_3)\in \bbfR^4$, for some $C>0$,\par (iii) either $b=1$, or $b=2$.\par The author obtains solutions having specified properties. In particular, the problem has infinitely many solutions.
[Anatolij Ivan Kolosov (Khar'kov)]
MSC 2000:
*34B15 Nonlinear boundary value problems of ODE

Keywords: fourth-order Sturm-Liouville problem; superlinear problem

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