Multiplicity of solutions for asymptotically linear \(n\)-th order boundary value problems (Q2475541)

The author is concerned with existence and multiplicity of nodal solutions of an \(n\)th order boundary value problem \[ u^{(n)}(t)+f(t,u(t))u(t)=0,\;t\in [0,\pi], \] \[ u^{(r)}(0)=0,\;r\in \{i_2,\dots, i_{n-1}\},\quad u(\pi)=u'(\pi)=0, \] where \(\{i_2,\cdots, i_{n-1}\}\) is a fixed set of distinct integers contained in \(\{0,\cdots, n-1\}, n\geq 3\) and \(f:[0,\pi]\times \mathbb{R}\rightarrow (-\infty, 0)\) is a continuous function. The proof follows a shooting approach and it is based on the weighted eigenvalue theory for linear \(n\)-order boundary value problems. For related work, see [\textit{R. Ma}, J. Math. Anal. Appl. 314, 254--265 (2006; Zbl 1085.34015)].