A singular Sturm-Liouville equation under non-homogeneous boundary conditions

Abstract

Given $\alpha > 0$ and $f\in L^2(0,1)$, consider the following singular Sturm-Liouville equation: \[ \left\lbrace\begin{aligned} -(x^{2\alpha}u'(x))'+u(x) & =f(x) \ \hbox{ a.e. on } (0,1),\\ u(1) & =0. \end{aligned}\right. \] We prove existence of solutions under (weighted) non-homogeneous boundary conditions at the origin.