On mixed local-nonlocal problems with Hardy potential

Abstract

In this paper we study the effect of the Hardy potential on existence, uniqueness and optimal summability of solutions of the mixed local-nonlocal elliptic problem $$-\Delta u + (-\Delta)^s u - \gamma \frac{u}{|x|^2}=f \text{ in } \Omega, \ u=0 \text{ in } \mathbb{R}^n \setminus \Omega,$$ where $\Omega$ is a bounded domain in $\mathbb{R}^n$ containing the origin and $\gamma> 0$. In particular, we will discuss the existence, non-existence and uniqueness of solutions in terms of the summability of $f$ and of the value of the parameter $\gamma$.