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.