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On certain nonlocal Hardy-Sobolev critical elliptic Dirichlet problems. (English) Zbl 1357.35283

Summary: 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 \[ \begin{cases} (-\Delta)^su-\gamma\frac u{|x|^{2s}}=\lambda u+\theta f(x,u)+ g(x,u)&\text{in}\;\Omega,\\ u=0&\text{in}\;\mathbb R^N\backslash\Omega, \end{cases} \] 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.”

MSC:

35R11 Fractional partial differential equations
35A15 Variational methods applied to PDEs
49J35 Existence of solutions for minimax problems
35B40 Asymptotic behavior of solutions to PDEs
35S15 Boundary value problems for PDEs with pseudodifferential operators
45G05 Singular nonlinear integral equations
47G20 Integro-differential operators
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Full Text: Euclid