Nonlinear eigenvalue problems having an unbounded branch of symmetric bound states (Q1575818)

The authors consider the nonlinear eigenvalue problem \[ u''(x)+\lambda u(x)+f(x,u(x))=0, x \in {\mathbb{R}}; \;\lim_{\mid x \mid \to \infty}{u(x)}=0, u(x) \geq 0; x \in {\mathbb{R}}; \;u \not \equiv 0. \] They prove the existence of a curve of positive solutions which bifurcates from the lowest eigenvalue of the linearized problem. This curve can be parametrized globally by \(\lambda\) and can be extended to infinity. They establish that all solutions to the equation are even and monotone and under appropriate conditions, all of them belong to the curve of bifurcation. The obtained results depend on the conditions of symmetry and monotonicity imposed to \(f\).