Infinitely many solutions of superlinear fourth order boundary value problems (Q701326)

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 (i) \(g:\mathbb{R}\to \mathbb{R}\) is continuous and satisfies \(\lim_{|\xi|\to\infty} g(\xi)/\xi= \infty\) (\(g\) is super-linear as \(|\xi|\to\infty\)), (ii) \(p: [0,1]\times \mathbb{R}^4\to\mathbb{R}\) is continuous and satisfies \(|p(x,\xi_0,\xi_1,\xi_2,\xi_3)|\leq C+{1\over 4}|\xi_0|\), \(x\in [0,1]\), \((\xi_0,\xi_1,\xi_2,\xi_3)\in \mathbb{R}^4\), for some \(C>0\), (iii) either \(b=1\), or \(b=2\). The author obtains solutions having specified properties. In particular, the problem has infinitely many solutions.