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 $$-\hat{I}”u+(-\hat{I}”{)}^{s}u-{\hat{I}}^3\frac{u}{|x{|}^2}=f\text{ in }\hat{I}\text{c◯},u=0\text{ in }{\mathbb{R}}^n\setminus \hat{I}\text{c◯},$$ where $\hat{I}\text{c◯}$ is a bounded domain in ${\mathbb{R}}^n$ containing the origin and ${\hat{I}}^3>0$. 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 ${\hat{I}}^3$.