On mixed local-nonlocal problems with Hardy potential

TL;DR

This paper investigates how the Hardy potential influences the existence, uniqueness, and integrability of solutions to a mixed local-nonlocal elliptic problem, highlighting the role of the parameter b3 and data summability.

Contribution

It provides new insights into the existence and uniqueness conditions for solutions of mixed local-nonlocal problems with Hardy potential, depending on data integrability and parameter values.

Findings

Existence of solutions depends on the summability of the source term and the Hardy parameter.

Non-existence results are established for certain parameter ranges.

Uniqueness of solutions is characterized in relation to the Hardy potential and data regularity.

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 $\gamma$.