$L^p$ theory for a singular Sturm-Liouville equation

Abstract: In this paper we consider the following Sturm-Liouville equation [ \left{ \begin{aligned} -(x^{2\alpha}u'(x))'+u(x)&=f(x) && \text{in } (0,1],\ u(1)&=0 \end{aligned} \right. ] where $\alpha<1$ is a nonzero real number and $f$ belongs to $L^p(0,1)$ for $p\geq 1$. We analyze the existence and regularity of solutions under suitable weighted Dirichlet boundary condition at the origin.