Choquard equation involving mixed local and nonlocal operators

TL;DR

This paper investigates an elliptic equation with mixed local and nonlocal operators, focusing on critical Hartree-type nonlinearity, establishing key inequalities, and proving existence and nonexistence results using variational methods.

Contribution

It introduces analysis of a novel elliptic problem with mixed operators and critical nonlinearity, including optimal inequality constants and solution criteria.

Findings

Established the Hardy-Littlewood-Sobolev inequality for the problem

Proved existence of solutions for subcritical cases

Demonstrated nonexistence under certain conditions

Abstract

In this article, we study an elliptic problem involving an operator of mixed order with both local and nonlocal aspects and in the presence of critical nonlinearity of Hartree type. To this end, we first investigate the corresponding Hardy-Littlewood-Sobolev inequality and detect the optimal constant. Using variational methods and a Poho\v{z}aev identity we then show the existence and nonexistence results for the corresponding subcritical perturbation problem.