New estimates for the Hardy constants of multipolar Schr\"odinger operators

Abstract

In this paper we study the optimization problem $$\mu^\star(\Omega):=\inf_{u\in \semi}\frac{\into |\n u|^2 \dx}{\into V u^2 \dx}$$ in a suitable functional space $\semi$. Here, $V$ is the multi-singular potential given by $$V:=\sum_{1\leq i<j\leq n} \frac{|a_i-a_j|^2}{|x-a_i|^2|x-a_j|^2}$$ and all the singular poles $a_1, \ldots, a_n$, $n\geq 2$, arise either in the interior or at the boundary of a smooth open domain $\Omega\subset \rr^N$, with $N\geq 3$ or $N \geq 2$, respectively. For a bounded domain $\Omega$ containing all the singularities in the interior, we prove that $\mu^\star(\Omega)>\mu^\star(\rr^N)$ when $n\geq 3$ and $\mu^\star(\Omega)=\mu^\star(\rr^N)$ when $n=2$ (It is known from \cite{cristi1} that $\mu^\star(\rr^N)=(N-2)^2/n^2)$. In the situation when all the poles are located on the boundary we show that $\mu^\star(\Omega)=N^2/n^2$ if $\Omega$ is either a ball, the exterior of a ball or a half-space. Our results do not depend on the distances between the poles. In addition, in the case of boundary singularities we obtain that $\mu^\star(\Omega)$ is attained in $\hoi$ when $\Omega$ is a ball and $n\geq 3$. Besides, $\mu^\star(\Omega)$ is attained in $\semi$ when $\Omega$ is the exterior of a ball with $N\geq 3$ and $n\geq 3$ whereas in the case of a half-space $\mu^\star(\Omega)$ is attained in $\semi$ when $n\geq 3$. We also analyze the critical constants in the so-called \textit{weak} Hardy inequality which characterizes the range of $\mu's$ ensuring the existence of a lower bound for the spectrum of the Schr\"{o}dinger operator $-\Delta -\mu V$. In the context of both interior and boundary singularities we show that the critical constants in the weak Hardy inequality are $(N-2)^2/(4n-4)$ and $N^2/(4n-4)$, respectively.