Elliptic operators with unbounded diffusion coefficients perturbed by inverse square potentials in $L^p$-spaces

Abstract

In this paper we give sufficient conditions on $\alpha \ge 0$ and $c\in \mathbb{R}$ ensuring that the space of test functions $C_c^{\infty }({\mathbb{R}}^N)$ is a core for the operator $$L_{0}u=(1+|x{|}^{\alpha })\Delta u+\frac{c}{|x{|}^2}u=:Lu+\frac{c}{|x{|}^2}u,$$ and $L_0$ with suitable domain generates a quasi-contractive and positivity preserving $C_0$-semigroup in $L^p({\mathbb{R}}^N),1<p<\infty$. The proofs are based on some $L^p$-weighted Hardy's inequality and perturbation techniques.

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