Advances in Differential Equations

On certain nonlocal Hardy-Sobolev critical elliptic Dirichlet problems

Alessio Fiscella and Patrizia Pucci

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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

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

First available in Project Euclid: 9 March 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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


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

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