On a singular boundary value problem for a second order ordinary differential equation
Consider the Dirichlet problem \[ u''+f(t,u,u')=0,\quad u(a)=A, u(b)=B, \] where \(f:(a,b)\times \mathbb R^2\to\mathbb R\) satisfies local Carathéodory conditions. Such conditions allow singularities at both end points \(a\) and \(b\). The authors define singular solutions, which are limit in \({\mathcal C}^1_{loc}(a,b)\) of approximate solutions. They prove that the existence of such a singular solution follows from the local Carathéodory conditions, an a-priori bound on the derivative and the existence of lower and upper solutions \(\alpha\leq\beta\). Classical solutions \(u\in{\mathcal C}([a,b])\cap W^{2,1}_{\text{loc}}(a,b)\) are obtained using controls on the nonlinearity \[ k(t)\leq f(t,u,u')\leq h(t) \] in the neighbourhood of both end points \(a\) and \(b\). This provides minimal and maximal solutions between the lower and upper solutions. An example of a singular equation that satisfies such controls is given.
scientific article; zbMATH DE number 1526168