Multiple positive solutions of Sturm-Liouville equations with singularities
The authors consider the second order Sturm-Liouville equation \[ (p(t) u'(t))' + g(t) F(t,u) = 0, \quad t \in (0, 1) \] with the (singular) boundary conditions \[ \alpha u(0) - \beta \lim_{t \rightarrow 0^+} p(t) u'(t) = 0, \gamma u(1) - \delta \lim_{t \rightarrow 1^-} p(t) u'(t) = 0, \] where \(\alpha, \beta, \gamma, \delta \geq 0\) and \[ \rho := \beta \gamma + \alpha \gamma \int_0^1 {dr \over p(r)} + \alpha \delta > 0. \] With an explicit representation of the Green's function of the corresponding homogeneous boundary value problem (where \(\rho\) is arisen), under a suitable assumption on \(p(t), g(t), F(t, u)\), by applying a fixed point theorem of the cone map, some existence and multiplicity results of positive solutions are obtained.
scientific article; zbMATH DE number 5134608