Infinitely many solutions of superlinear fourth order boundary value problems

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:
\begin{itemize}
\item[(i)]
$g \colon \mathbb R \to \mathbb R$ is continuous and
satisfies
$\lim_{|\xi| \to \infty} g(\xi)/\xi =\infty$


($g$ is
< i> superlinear< /i> as $|\xi| \to \infty$),
\item[(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 $C> 0$,
\item[(iii)]
either $b=1$ or $b=2$.
\end{itemize}
We obtain solutions having specified nodal properties.
In particular, the problem has infinitely many solutions.