Some results on second-order elliptic operators with polynomially growing coefficients in $L^p$-spaces

Abstract

In this paper we study minimal realizations in $L^p({\mathbb{R}}^N)$ of the second order elliptic operator \begin{equation*} { A_{b,c}} := (1+|x|^α)Δ+ b|x|^{α-2}x\cdot\nabla - c |x|^{α-2} - |x|^β , \quad x \in \mathbb{R}^N, \end{equation*} where $N\ge 3$, $\alpha \in [0,2)$, $\beta >0$, and $b,c$ are real numbers. We use quadratic form methods to prove that $\left(A_{b,c},C_c^{\infty }\left({\mathbb{R}}^N\setminus \{0\}\right)\right)$ admits an extension that generates an analytic $C_0-$semigroup for all $p\in (1,\infty )$. Moreover, we give conditions on the coefficients under which this extension is precisely the closure of $\left(A_{b,c},C_c^{\infty }\left({\mathbb{R}}^N\setminus \{0\}\right)\right)$.