The structure of the solution set of a class of nonlinear eigenvalue problems
The author considers the nonlinear eigenvalue problem \[ x=f (\lambda,x) \quad \text{for } 0\leq \lambda<\infty, \tag{1} \] \(x\) in a positive cone of an ordered Banach space, \(f\) completely continuous with \(f(\lambda,0) \neq 0\) for \(\lambda >0\). Let \(\Sigma\) be the set of solutions of (1), \(\Lambda\) be the set of all \(\lambda\) for which \((\lambda,x) \in \Sigma\) for some \(x\), and \(\lambda^*=\sup \Lambda\). He cites theorems due to Amann and to Dancer. The first says that, under certain hypotheses \(H_ 1\) on \(f:\) \(\Lambda\) is an interval containing 0; for every \(\lambda \in \Lambda\), (1) has a minimal solution \(\overline x(\lambda)\) which is left continuous and strongly increasing; if \(\lambda^*<\infty\), then \(\lambda^*\in \Lambda\) iff \(\bigl\{ \overline x(\lambda):0 \leq \lambda<\lambda^*\bigr\}\) is a bounded set. The second says that, under certain hypotheses \(H_ 2\) on \(f\), the component \(S_ 0\) of \(\Sigma\) which contains (0,0) is unbounded. The author then states: ``The purpose of this paper is to give a detailed description of the structure of \(S_ 0\) when \(f\) satisfies \(H_ 1\) and \(H_ 2\) simultaneously. In particular, we prove that all the minimal solutions \(\biggl\{ \bigl( \lambda,\overline x(\lambda) \bigr) :0 \leq \lambda<\lambda^*\biggr\}\) are contained in \(S_ 0\), and that if \(\lambda^*\in \Lambda\) (this is the most interesting case, many authors give conditions so that \(\lambda^*\in \Lambda)\), then \(S_ 0\) is composed of two connected sets \(c_ 0\) and \(c_ \infty\), \(c_ 0\) goes from \(\bigl( \lambda^*,\overline x(\lambda^*) \bigr)\) through \(\biggl\{ \bigl( \lambda,\overline x (\lambda) \bigr) :0 \leq \lambda<\lambda^* \biggr\}\) to (0,0) and \(c_ \infty\) from \(\bigl( \lambda^*, \overline x(\lambda^*) \bigr)\) to infinity, \(c_ 0\cap c_ \infty= \bigl( \lambda^*, \overline x(\lambda^*) \bigr)\).'' He applies his results to \[ \Delta u=\lambda \varphi(x,u) \] for \(x\) in a bounded domain of \(\mathbb{R}^ n\), \(u=0\) on its boundary.
scientific article; zbMATH DE number 150671
scientific article