Forced secondary bifurcation in an elliptic boundary value problem
In this paper, the author studies the problem \[ -\Delta u-\lambda u+u^ 3=h \quad\text{in}\quad \Omega, \qquad \partial u/\partial n=0 \quad\text{on} \quad \partial \Omega, \] where \(0<\lambda<\lambda_ 2\) and \(\lambda_ 2\) denotes the second eigenvalue of \(-\Delta\) on \(\Omega\) under Neumann boundary conditions. In earlier work he proved that if \(\lambda<{1\over 12}\lambda_ 2\) then for each \(h\) there are at most 3 solutions. In fact if \(\lambda\) is small enough, but positive it can be proved that his map is topologically equivalent to a cusp map. (cf. the work of the reviewer, \textit{P. T. Church} and \textit{J. G. Timourian} [The structure of a nonlinear elliptic operator, preprint (1989)] for a related result.) Here he shows that, if \(\Omega=(0,1)\) and \({2\lambda_ 2 \over \pi^ 2}<\lambda<\lambda_ 2\), there is an \(h\) for which there are more than three solutions. Thus the expected generalization of his original result is false.
semi-linear elliptic boundary value problem
scientific article; zbMATH DE number 90505