A semilinear singular Sturm–Liouville equation involving measure data

Abstract

Given $\alpha >0$ and $p>1$, let μ be a bounded Radon measure on the interval $(-1,1)$. We are interested in the equation $-(|x{|}^{2\alpha }{u}^{\prime }{)}^{\prime }+|u{|}^{p-1}u=\mu$ on $(-1,1)$ with boundary condition $u(-1)=u(1)=0$. We establish some existence and uniqueness results. We examine the limiting behavior of three approximation schemes. The isolated singularity at 0 is also investigated.