Some results on the existence and multiplicity of Dirichlet type solutions for a singular equation coming from a Kepler problem on the sphere

We study the Dirichlet boundary value problem u′′=h(t)sin2u,u(0+)=c1,u(T−)=c2,where c1,c2∈[0,π] and h:[0,T]→R is a Lebesgue integrable function. The forcing term under consideration is the product of a nonlinearity which is singular at two points with a weight function h. We prove that the corresponding singular boundary value problem is solvable only if the weight function does not change its sign. Therefore, our main result is stated under this setting: supposing that h:[0,T]→[0,+∞), the existence and multiplicity of solutions to the aforementioned problem is guaranteed if and only if h¯ is small enough.