Symmetry-breaking bifurcation for the one-dimensional Hénon equation

Abstract

We show the existence of a symmetry-breaking bifurcation point for the one-dimensional Hénon equation [Formula: see text] where [Formula: see text] and [Formula: see text]. 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 [Formula: see text] where [Formula: see text] is a specified continuous parametrization function.

Symmetry-breaking bifurcation for the Moore–Nehari differential equation

“…It is well known that the problem above has a unique solution and it becomes even (for example, see [8] or [17]). Clearly, this solution U (x) is written as U (x) = y(x + 1), where y(x) is given by (2.12).…”

“…Kajikiya [6,7] proved that a non-even solution given in Theorem 1.1 can be obtained as a least energy solution of R(u) in which |x| λ is replaced by h(x, λ). Sim and Tanaka [17] studied (1.2) when N = 1, i.e., u + |x| λ u p = 0, u > 0 in (−1, 1), u(−1) = u(1) = 0.…”

Secondary bifurcations in semilinear ordinary differential equations

“…In many such problems, secondary bifurcations are observed. For instance, the Neumann problem for the equation (c(x)u x ) x + λ(u − u 3 ) = 0 with a piecewise constant function c(x) was studied by [10], and the Dirichlet problem for the equation u xx + d(λ, x, u) = 0 with d(λ, x, u) = |x| λ u p , |x| λ e u , or (1 − χ (−λ,λ) (x))u p was investigated by [12,21,23,24]. We remark that the last two conditions of (1.1) also appear when we consider the Schrödinger operators with δ -interactions (for recent studies see [1][2][3][4]9]).…”