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.
Citation
Bryan P. Rynne. "Infinitely many solutions of superlinear fourth order boundary value problems." Topol. Methods Nonlinear Anal. 19 (2) 303 - 312, 2002.
Information