# Mixed local and nonlocal semilinear elliptic equation with strongly singular and critical Choquard nonlinearity

## TL;DR

This paper investigates a complex elliptic equation combining local and nonlocal features, singular and critical nonlinearities, establishing existence and multiplicity of positive solutions using advanced variational and nonsmooth analysis techniques.

## Contribution

It introduces a novel approach to handle mixed local and nonlocal elliptic problems with singular and critical nonlinearities, proving new existence and multiplicity results.

## Key findings

- Existence of positive solutions for certain parameter ranges
- Multiple solutions under specific conditions
- Application of nonsmooth critical point theory to complex nonlinear problems

## Abstract

In this article, we study an elliptic problem of mixed order with both local and nonlocal aspects involving singular nonlinearity in combination with critical Hartree-type nonlinearity. Using variational methods together with the critical point theory of nonsmooth analysis and the geometry of the energy functional, we show the existence and multiplicity of positive solutions with respect to the parameter $\lambda$.