## Topological Methods in Nonlinear Analysis

### Infinitely many solutions of superlinear fourth order boundary value problems

Bryan P. Rynne

#### Abstract

We consider the boundary value problem \begin{gather*} 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, \end{gather*} where:

(i) $g \colon \mathbb R \to \mathbb R$ is continuous and satisfies $\lim_{|\xi| \to \infty} g(\xi)/\xi =\infty$ ($g$ is superlinear as $|\xi| \to \infty$),

(ii) $p \colon [0,1] \times \mathbb R^4 \to \mathbb R$ is continuous and satisfies $$|p(x,\xi_0,\xi_1,\xi_2,\xi_3)| \le C + \frac{1}{4} |\xi_0| , \quad x \in [0,1],\ (\xi_0,\xi_1,\xi_2,\xi_3) \in \mathbb R^4,$$ for some $C> 0$,

(iii) either $b=1$ or $b=2$.

We obtain solutions having specified nodal properties. In particular, the problem has infinitely many solutions.

#### Article information

Source
Topol. Methods Nonlinear Anal., Volume 19, Number 2 (2002), 303-312.

Dates
First available in Project Euclid: 2 August 2016

https://projecteuclid.org/euclid.tmna/1470138766

Mathematical Reviews number (MathSciNet)
MR1921051

Zentralblatt MATH identifier
1017.34015

#### Citation

Rynne, Bryan P. Infinitely many solutions of superlinear fourth order boundary value problems. Topol. Methods Nonlinear Anal. 19 (2002), no. 2, 303--312. https://projecteuclid.org/euclid.tmna/1470138766

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