## Advances in Differential Equations

- Adv. Differential Equations
- Volume 21, Number 5/6 (2016), 571-599.

### On certain nonlocal Hardy-Sobolev critical elliptic Dirichlet problems

Alessio Fiscella and Patrizia Pucci

#### Abstract

This paper deals with the existence, multiplicity and the asymptotic behavior of nontrivial solutions for nonlinear problems driven by the fractional Laplace operator $(-\Delta)^s$ and involving a critical Hardy potential. In particular, we consider $$ \left\{ \begin{array}{ll} (- \Delta)^{s}u - \gamma \displaystyle \frac{u}{|x|^{2s}} = \lambda u + \theta f(x,u) +g(x,u) & \mbox{ in }\Omega,\\ u=0 & \mbox{in} \mathbb{R}^{N} \setminus \Omega, \end{array} \right. $$ where $\Omega\subset \mathbb R^N$ is a bounded domain, $\gamma, \lambda$ and $\theta$ are real parameters, the function $f$ is a subcritical nonlinearity, while $g$ could be either a critical term or a perturbation.

#### Article information

**Source**

Adv. Differential Equations, Volume 21, Number 5/6 (2016), 571-599.

**Dates**

First available in Project Euclid: 9 March 2016

**Permanent link to this document**

https://projecteuclid.org/euclid.ade/1457536500

**Mathematical Reviews number (MathSciNet)**

MR3473584

**Zentralblatt MATH identifier**

1357.35283

**Subjects**

Primary: 49J35: Minimax problems 35A15: Variational methods 35S15: Boundary value problems for pseudodifferential operators 47G20: Integro-differential operators [See also 34K30, 35R09, 35R10, 45Jxx, 45Kxx] 45G05: Singular nonlinear integral equations

#### Citation

Fiscella, Alessio; Pucci, Patrizia. On certain nonlocal Hardy-Sobolev critical elliptic Dirichlet problems. Adv. Differential Equations 21 (2016), no. 5/6, 571--599. https://projecteuclid.org/euclid.ade/1457536500