Global solution curves for boundary value problems, with linear part at resonance
The exact number of solutions is obtained for both the Dirichlet problem \[ \begin{cases} u''(x)+u(x)+kg(u(x))=\mu \sin x +e(x) \;\;\text{ for } x\in (0, \pi),\\ u(0)=u(\pi)=0\end{cases} \tag{1} \] and the Neumann problem \[ \begin{cases} u''(x)+kg(u(x))=\mu +e(x) \;\;\text{ for } x\in (0, \pi),\\ u'(0)=u'(\pi)=0\end{cases}\tag{2} \] by a dynamical version of the classical Liapunov-Schmidt procedure. Moreover, the paper performs continuation in the subspace orthogonal to the kernel after decomposing the space. The main results include uniqueness and infinite multiplicity. Theorem 4.1, one of the main results is as follows: Assume that \(g \in C^{1}(\mathbb R)\) in (1) satisfies \(|g(u)|\leq M_{0}\) for all \(u \in\mathbb R\) , \(|g'(u)|\leq M \) for all \(u \in\mathbb R\) \ with some positive constants \(M_{0}\)\ and \(M\). Assume that \(k\leq \frac{3}{M},\;e\in L^{2}_{1} \) is a given function with \( \int^{\pi}_{0}e(x)\sin xdx=0\). In addition, assume that \(\mu\) satisfies the necessary condition \(\frac{4}{\pi}kg(-\infty)<\mu<\frac{4}{\pi}kg(\infty)\), \ where \(g(\infty)=\lim_{u\to \infty}g(u)\)\ and \(g(-\infty)=\lim_{u\to -\infty}g(u)\). Then problem (1) has a solution. Moreover, all solutions of (1) lie on a unique solution curve, which is given by an equation \(\mu = \phi (\xi)\), where \(\phi(\xi)\) is a bounded continuous function, having limits at infinity: \(\phi (-\infty)=\frac{4}{\pi}kg(-\infty)\) and \(\phi (\infty)=\frac{4}{\pi}kg(\infty)\).
scientific article; zbMATH DE number 5586782
scientific article