Existence and multiplicity of solutions for Hardy nonlocal fractional elliptic equations involving critical nonlinearities

  • Somayeh Rastegarzadeh
  • Nemat NyamoradiEmail author
  • Vincenzo Ambrosio


In this paper, systems of fractional Laplacian equations are investigated, which involve critical homogeneous nonlinearities and Hardy-type terms as follows
$$\begin{aligned} {\left\{ \begin{array}{ll} &{}(- \Delta )^s u - \gamma \frac{u}{|x|^{2 s}} = \frac{\lambda \mu }{\mu + \eta } \frac{|u|^{\mu - 2} |v|^{\eta } u}{|x|^{\alpha }}+ \frac{Q(x)}{2^*_s (\alpha )}\frac{ H_u(u,v)}{|x|^{\alpha }}, ~ in~ {\mathbb {R}}^N,\\ \\ &{} (- \Delta )^s v - \gamma \frac{v}{|x|^{2 s}} = \frac{\lambda \eta }{\mu + \eta } \frac{|u|^{\mu } |v|^{\eta - 2} v}{|x|^{\alpha }} + \frac{Q(x)}{2^*_s (\alpha )}\frac{ H_v(u,v)}{|x|^{\alpha }}, ~ in ~{\mathbb {R}}^N, \end{array}\right. } \end{aligned}$$
where \(0< s < 1\), \(0< \alpha< 2s <N\), \(0< \gamma < \gamma _H\) with
$$\begin{aligned} \gamma _H =4^s \frac{\Gamma ^2(\frac{N+2s}{4})}{\Gamma ^2(\frac{N-2s}{4})}, \;\;\; \text {for} \; i = 1,\ldots ,l \end{aligned}$$
being the fractional best Hardy constant on \({\mathbb {R}}^N\), \(2^*_s (\alpha ) = \frac{2 (N -\alpha )}{N - 2 s}\) is critical Hardy–Sobolev exponent, \(\mu , \eta > 1\) with \(\mu + \eta = 2^*_s(\alpha )\), Q is G-symmetric functions (G is a closed subgroup of O(N), see Sect. 2 for details) satisfying some appropriate conditions which will be specified later, \(\lambda \) is real parameter, \(H_u\), \(H_v\) are the partial derivatives of the 2-variable \(C^1\)-functions H(uv) and \((-\Delta )^s\) is the fractional Laplacian operator which (up to normalization factors) may be defined as
$$\begin{aligned} (- \Delta )^s u (x) = - \frac{1}{2}\int _ {{\mathbb {R}}^{N} } \frac{u (x + y) + u (x - y) - 2 u (x) }{|x-y|^{N + 2 s}}{\text {d}}y. \end{aligned}$$
By variational methods and local compactness of Palais–Smale sequences, the extremals of the corresponding best Hardy–Sobolev constant are found and the existence of solutions to the system is established.


Fractional Laplacian solution Hardy–Sobolev exponent mountain-pass theorem concentration compactness principle 

Mathematics Subject Classification

35R11 35A15 35J60 47G20 35J20 



The authors would like to thank the referees for their suggestions and helpful comments which improved the presentation of the original manuscript.


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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Somayeh Rastegarzadeh
    • 1
  • Nemat Nyamoradi
    • 1
    Email author
  • Vincenzo Ambrosio
    • 2
  1. 1.Department of MathematicsFaculty of Sciences, Razi UniversityKermanshahIran
  2. 2.Dipartimento di Ingegneria Industriale e Scienze MatematicheUniversità Politecnica delle MarcheAnconaItaly

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