Symmetry-breaking bifurcation for the one-dimensional Liouville type equation

The two-point boundary value problem for the one-dimensional Liouville type equation{u″+λ|x|leu=0,x∈(−1,1),u(−1)=u(1)=0 is considered, where λ>0 and l>0. In this paper, a symmetry-breaking result is obtained by using the Morse index. The problem{u″+λ|x|l(u+1)p=0,x∈(−1,1),u(−1)=u(1)=0 is also considered, where λ>0, l>0, p>1 and (p−1)l>4.