Secondary bifurcations in semilinear ordinary differential equations

We consider the Neumann problem for the equation $$u_{xx}+{\uplambda }f(u)=0$$ u xx + λ f ( u ) = 0 in the punctured interval $$(-1,1) \setminus \{0\}$$ ( - 1 , 1 ) \ { 0 } , where $${\uplambda }>0$$ λ > 0 is a bifurcation parameter and $$f(u)=u-u^3$$ f ( u ) = u - u 3 . At $$x=0$$ x = 0 , we impose the conditions $$u(-0)+au_x(-0)=u(+0)-au_x(+0)$$ u ( - 0 ) + a u x ( - 0 ) = u ( + 0 ) - a u x ( + 0 ) and $$u_x(-0)=u_x(+0)$$ u x ( - 0 ) = u x ( + 0 ) for a constant $$a>0$$ a > 0 (the symbols $$+0$$ + 0 and $$-0$$ - 0 stand for one-sided limits). The problem appears as a limiting equation for a semilinear elliptic equation in a higher dimensional domain shrinking to the interval $$(-1,1)$$ ( - 1 , 1 ) . First we prove that odd solutions and even solutions form families of branches $$\{ {{\mathcal {C}}}^o_k\}_{k \in {{\mathbb {N}}}}$$ { C k o } k ∈ N and $$\{ {{\mathcal {C}}}^e_k\}_{k \in {{\mathbb {N}}}}$$ { C k e } k ∈ N , respectively. Both $${{\mathcal {C}}}^o_k$$ C k o and $${{\mathcal {C}}}^e_k$$ C k e bifurcate from the trivial solution $$u=0$$ u = 0 . We then show that $${{\mathcal {C}}}^e_k$$ C k e contains no other bifurcation point, while $${{\mathcal {C}}}^o_k$$ C k o contains two points where secondary bifurcations occur. Finally we determine the Morse index of solutions on the branches. General conditions on f(u) for the same assertions to hold are also given.