# A Global Compact Result for a Fractional Elliptic Problem with Hardy term and critical non-linearity on the whole space

## TL;DR

This paper establishes the existence of positive solutions for a fractional elliptic equation with Hardy potential and critical Sobolev nonlinearity on the entire space, using compactness analysis of the associated functional.

## Contribution

It provides a new existence result for fractional elliptic equations with Hardy terms and critical nonlinearity, under specific conditions on the coefficients.

## Key findings

- Existence of positive solutions under certain assumptions.
- Compactness analysis of the associated functional.
- Extension of results to fractional elliptic equations with Hardy potential.

## Abstract

In this paper, we deal with a fractional elliptic equation with critical Sobolev nonlinearity and Hardy term $$ (-\Delta)^{\alpha} u-\mu\frac{u}{|x|^{2\alpha}}+a(x) u=|u|^{2^*-2}u+k(x)|u|^{q-2}u$$ $$ u\,\in\,H^\alpha({\mathbb R}^N),$$ where $2<q< 2^*$, $0<\alpha<1$, $N>4\alpha$, $2^*=2N/(N-2\alpha)$ is the critical Sobolev exponent, $a(x),k(x)\in C({\mathbb R}^N)$. Through a compactness analysis of the functional associated to $(*)$, we obtain the existence of positive solutions for $(*)$ under certain assumptions on $a(x),k(x)$.