Sharp $H^2$-regularity in dimensions $N\geq 5$ and beyond for two classes of elliptic problems with critical unbounded coefficients

Abstract

We establish sharp parameter thresholds governing $H^2$ -regularity of weak solutions in $H_0^1$ for two classes of elliptic problems with critical unbounded perturbations, in dimensions $N\geq 5$. More precisely, we consider two distinct $\lambda$-parametric elliptic problems $ - \Delta v + \lambda \frac{x\cdot \nabla v}{|x|^{2}} =f$ and $-\Delta v + \lambda \frac{v}{|x|^2}=f$ posed in a bounded $C^2$-domain $\Omega\subset \mathbb{R}^N$ containing the origin $x=0$. We observe that the singular perturbations $\frac{x\cdot \nabla v}{|x|^{2}}$ and $\frac{v}{|x|^2}$ are homogeneous operators of order 2 consistent with the scaling of the Laplacian. In view of the Hardy inequality the problems are well-posed in $H_0^1(\Omega)$ for $\lambda<\frac{N-2}{2}$ and $\lambda>-\frac{(N-2)^2}{4}$ respectively. The main results are as follows. For the first problem we show that any solution $v\in H_0^1(\Omega)$ belongs to $H^2(\Omega)$ for any $\lambda< \frac{N-2}{2}$ provided $f\in L^2(\Omega)$. This fully extends the previous $H^2$ regularity properties obtained by Kim and Tsai in \cite{Kim-Tsai} for $\lambda\leq 0$. For the second problem we show that $H^2$ regularity holds for any $\lambda>- \frac{N(N-4)}{4}$ and fails for any $\lambda\in \left(-\frac{(N-2)^2}{4},-\frac{N(N-4)}{4}\right]$. This extends sharply the range of $\lambda \in \left(-\frac{N(N-4)}{4}, \frac{N(N-4)}{4}\right)$ obtained when applying the Kato perturbation theory in \cite{Kato}. In addition, we develop sharp second order Hardy-Rellich type inequalities for the involved elliptic operators which are essential in the above proofs.