On mixed local-nonlocal problems with Hardy potential

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 $$-\mathrm{\Delta }u+(-\mathrm{\Delta }{)}^{s}u-\gamma \frac{u}{\mathrm{\mid }x{\mathrm{\mid }}^2}=f\text{ in }\mathrm{\Omega },u=0\text{ in }{\mathbb{R}}^n\setminus \mathrm{\Omega },$$ where $\mathrm{\Omega }$ 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 $\gamma$.