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

TL;DR

This paper develops an $L^p$ theory for a singular Sturm-Liouville problem involving a weighted differential operator, establishing existence and regularity of solutions for functions in $L^p$ spaces.

Contribution

It introduces a novel $L^p$ framework for analyzing solutions to a singular Sturm-Liouville equation with weighted derivatives and boundary conditions.

Findings

Existence of solutions in weighted $L^p$ spaces.

Regularity results for solutions near the singularity.

Extension of classical Sturm-Liouville theory to singular weights.

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.