Bifurcation of sign-changing solutions for \(m\)-point boundary value problems
From the introduction: We consider the \(m\)-point boundary value problems \[ u''+\lambda f(u)= 0,\quad t\in (0,1),\tag{1} \] \[ u(0)= 0,\quad u(1)= \sum^{m-2}_{i=1} \alpha_i u(\eta_i),\tag{2} \] where integer \(m\geq 3\), \(\eta_i\in (0,1)\), \(\lambda\) is a positive parameter, and \(f\in C^1(\mathbb{R},\mathbb{R})\). We study the multiplicity and global structure of sign-changing solutions of (1) and (2) under the assumptions (A1) \(\alpha_i>0\) for \(i= 1,\dots, m-2\) with \(0< \sum^{m-2}_{i=1} \alpha_i< 1\); (A2) \(f\in C^1(\mathbb{R},\mathbb{R})\) satisfies \(sf(s)> 0\) for \(s\neq 0\); (A3) \(f_0:= \lim_{|s|\to 0} f(s)/s= 0\); (A4) \(f_\infty:= \lim_{|x|\to\infty} f(s)/s= 0\).
scientific article; zbMATH DE number 6049784
scientific article