Nonlinear eigenvalue problems having an unbounded branch of symmetric bound states

Abstract

For a semilinear second order differential equation on $\mathbb{R}$, the existence of a continuous curve of positive solutions which bifurcates from the lowest eigenvalue of the linearized problem is proved. This curve can be parameterized globally by $\lambda$ and can be extended to infinity. We establish that all solutions of the equation are even and monotone, and under appropriate conditions, all of them belong to the curve of bifurcation. Our results depend heavily on the combination of symmetry and monotonicity imposed on the equation.