Symmetry-breaking bifurcation for the ono-dimensional Henon equation

We show the existence of a symmetry-breaking bifurcation point for the one-dimensional Henon equation u '' + vertical bar x vertical bar(l)u(p )= 0, x is an element of (-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 symmetrybreaking bifurcation points (the second of the alternatives about Rabinowitz's global bifurcation) for the problem u '' + vertical bar x vertical bar(l(lambda))u(p )= 0, x is an element of (-1, 1), u(-1) = u(1) = 0, where l is a specified continuous parametrization function.