Finite many critical problems involving Fractional Laplacians in $\mathbb{R}^{N}$

TL;DR

This paper investigates nonlocal elliptic problems with multiple critical exponents involving fractional Laplacians in Euclidean space, establishing existence results using advanced inequalities and variational methods.

Contribution

It introduces new existence results for fractional Laplacian problems with multiple critical exponents, extending previous work by employing refined inequalities and fractional Coulomb--Sobolev spaces.

Findings

Existence of nonnegative solutions established.

Generalization of previous results by Yang and Wu.

Application of endpoint refined Hardy--Sobolev inequality.

Abstract

In this paper, we consider the nonlocal elliptic problems in $\mathbb{R}^{N}$, which involve finite many critical exponents. By using endpoint refined Hardy--Sobolev inequality, fractional Coulomb--Sobolev space and variational method, we establish the existence of nonnegative solution. Our results generalize some results obtained by Yang and Wu [Adv. Nonlinear Stud. (2017) \cite{Yang2017}].