Some results on the existence and multiplicity of Dirichlet type solutions for a singular equation coming from a Kepler problem on the sphere. (February 2019) - Godoy, José; Zamora, Manuel
Abstract: We study the Dirichlet boundary value problem u ′ ′ = h ( t ) sin 2 u, u ( 0 + ) = c 1, u ( T − ) = c 2, where c 1, c 2 ∈ [ 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.