Symmetry-breaking bifurcation for the one-dimensional Hénon equation

We show the existence of a symmetry-breaking bifurcation point for the one-dimensional Hénon equation u″ + |x|^lu^p = 0, x ∈ (-1, 1), u(-1) = u(1) = 0, where l > 0 and p > 1. Moreover, employing a variant of Rabinowitz's global bifurcation, we obtain the unbounded connected set (the first of the alternatives about Rabinowitz's global bifurcation), which emanates from the symmetry-breaking bifurcation point. Finally, we give an example of a bounded branch connecting two symmetry-breaking bifurcation points (the second of the alternatives about Rabinowitz's global bifurcation) for the problem u″ + |x|^l(λ)u^p = 0, x ∈ (-1, 1), u(-1) = u(1) = 0, where l is a specified continuous parametrization function.