The Brezis-Nirenberg Result for the Fractional Elliptic Problem with Singular Potential

TL;DR

This paper establishes the existence of solutions for a fractional elliptic problem with singular potential, extending the classical Brezis-Nirenberg result to fractional operators with critical Sobolev exponent.

Contribution

It introduces a variational approach to solve fractional elliptic equations with singular potentials and derives a Brezis-Nirenberg type result in this context.

Findings

Existence of solutions for the fractional problem with singular potential.

Extension of Brezis-Nirenberg result to fractional operators.

Application of variational methods to critical Sobolev problems.

Abstract

In this paper, we are concerned with the following type of fractional problems: $$ \begin{cases}\dis (-\Delta)^{s} u-\mu\frac{u}{|x|^{2s}}-\lambda u=|u|^{2^*_{s}-2}u+f(x,u), &\text{in} \Omega,\ \ \, u=0\,&\text{in} \R^N\backslash\Omega \end{cases} \eqno {(*)} $$ where $s\in (0,1)$, $2^*_{s}=2N/(N-2s)$ is the critical Sobolev exponent, $f(x,u)$ is a lower order perturbation of critical Sobolev nonlinearity. We obtain the existence of the solution for (*) through variational methods. In particular we derive a Br\'ezis-Nirenberg type result when $f(x,u)=0$.