## Topological Methods in Nonlinear Analysis

### Existence of positive solutions for Hardy nonlocal fractional elliptic equations involving critical nonlinearities

#### Abstract

In this paper, we have used variational methods to study existence of solutions for the following critical nonlocal fractional Hardy elliptic equation \begin{equation*} (- \Delta)^s u - \gamma \frac{u}{|x|^{2 s}} = \frac{|u|^{2_s^*(b) - 2} u}{|x|^{b}} + \lambda f (x, u ), \quad \text{in } \mathbb{R}^N, \end{equation*} where $N > 2 s$, $0< s< 1$, $\gamma, \lambda$ are real parameters, $(- \Delta)^s$ is the fractional Laplace operator, $2_s^*(b) = {2 (N - b)}/(N - 2s)$ is a critical Hardy-Sobolev exponent with $b \in [0, 2s)$ and $f \in C(\mathbb R^{N} \times \mathbb{R}, \mathbb{R})$.

#### Article information

Source
Topol. Methods Nonlinear Anal., Volume 53, Number 2 (2019), 731-746.

Dates
First available in Project Euclid: 11 May 2019

https://projecteuclid.org/euclid.tmna/1557540145

Digital Object Identifier
doi:10.12775/TMNA.2019.021

Mathematical Reviews number (MathSciNet)
MR3983992

#### Citation

Rastegarzadeh, Somayeh; Nyamoradi, Nemat. Existence of positive solutions for Hardy nonlocal fractional elliptic equations involving critical nonlinearities. Topol. Methods Nonlinear Anal. 53 (2019), no. 2, 731--746. doi:10.12775/TMNA.2019.021. https://projecteuclid.org/euclid.tmna/1557540145

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