Multiplicity and eigenvalue intersecting nonlinearities (Q913137)

In a bounded domain \(\Omega \subset {\mathbb{R}}^ N\), nonlinear elliptic equations \[ (*)\quad -\Delta u=f(u)+h\quad in\quad \Omega,\quad u=0\quad on\quad \partial \Omega \] are considered, where f is continuous and where f(s)/s has limits \(f^+\) and \(f^-\) in \({\mathbb{R}}\) as \(s\to \pm \infty\), respectively. Firstly, the author reviews results on the minimal number of solutions to (*), provided that some of the eigenvalues \(\lambda_ k\) of the Laplace operator \(\Delta\) satisfy \(f^+>\lambda_ k>f^-\) or \(f^+<\lambda_ k<f^-\). Then, for the one-dimensional problem, additional results are presented.