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

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