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

  • Somayeh Rastegarzadeh
  • Nemat NyamoradiEmail author
  • Vincenzo Ambrosio
Article
  • 35 Downloads

Abstract

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.

Keywords

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

Mathematics Subject Classification

35R11 35A15 35J60 47G20 35J20 

Notes

Acknowledgements

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

References

  1. 1.
    Abdellaoui, B., Bentifour, R.: Caffarelli–Kohn–Nirenberg type inequalities of fractional order with applications. J. Funct. Anal. 272(10), 3998–4029 (2017)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Ambrosetti, A., Rabinowitz, P.H.: Dual variational methods in critical point theory and applications. J. Funct. Anal. 14, 349–381 (1973)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Ambrosio, V.: Ground states solutions for a non-linear equation involving a pseudo-relativistic Schrödinger operator. J. Math. Phys. 57(5), 051502 (2016)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Ambrosio, V., Hajaiej, H.: Multiple solutions for a class of nonhomogeneous fractional Schrödinger equations in \({\mathbb{R}}^{N}\). J. Dynam. Differ. Equ. 30(3), 1119–1143 (2018)CrossRefGoogle Scholar
  5. 5.
    Ambrosio, V.: Periodic solutions for critical fractional equations. Calc. Var. Partial Differ. Equ 57(2), 45 (2018)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Ambrosio, V.: Concentration phenomena for critical fractional Schrödinger systems. Commun. Pure Appl. Anal. 17(5), 2085–2123 (2018)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Applebaum, D.: Lévy processes-from probability to finance and quantum groups. Notices Am. Math. Soc. 51(11), 1336–1347 (2004)zbMATHGoogle Scholar
  8. 8.
    Barrios, B., Medina, M., Peral, I.: Some remarks on the solvability of non-local elliptic problems with the Hardy potential. Commun. Contemp. Math. 16(4), 1350046 (2014)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Bianchi, G., Chabrowski, J., Szulkin, A.: On symmetric solutions of an elliptic equation with a nonlinearity involving critical Sobolev exponent. Nonlinear Anal. 25(1), 41–59 (1995)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Caffarelli, L.: Nonlinear partial differential equations. Abel Symposia. 7, 37–52 (2012)CrossRefGoogle Scholar
  11. 11.
    Caffarelli, L., Silvestre, L.: An extension problem related to the fractional Laplacian. Commun. Partial Differ. Equ. 32(7–9), 1245–1260 (2007)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Caponi, M., Pucci, P.: Existence theorems for entire solutions of stationary Kirchhoff fractional \(p\)-Laplacian equations. Ann. Mat. Pura Appl. 195(6), 2099–2129 (2016)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Chen, W.: Fractional elliptic problems with two critical Sobolev–Hardy exponents. Electron. J. Differ. Equ. 2018(22), 1–12 (2018)Google Scholar
  14. 14.
    de Morais Filho, D.C., Souto, M.A.S.: Systems of \(p\)-Laplacian equations involving homogeneous nonlinearities with critical Sobolev exponent degrees. Commun. Partial Differ. Equ. 24(7–8), 1537–1553 (1999)CrossRefGoogle Scholar
  15. 15.
    Deng, Y.B., Jin, L.Y.: On symmetric solutions of a singular elliptic equation with critical Sobolev–Hardy exponent. J. Math. Anal. Appl. 329(1), 603–616 (2007)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Deng, Z., Huang, Y.: On \(G\)-symmetric solutions of a quasilinear elliptic equation involving critical Hardy–Sobolev exponent. J. Math. Anal. Appl. 384, 578–590 (2011)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Deng, Z., Huang, Y.: Existence of symmetric solutions for singular semilinear elliptic systems with critical Hardy–Sobolev exponents. Nonlinear Anal. Real World Appl. 14(1), 613–625 (2013)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Di Nezza, E., Palatucci, G., Valdinoci, E.: Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136(5), 521–573 (2012)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Dipierro, S., Medina, M., Valdinoci, E.: Fractional elliptic problems with critical growth in the whole of \({\mathbb{R}}^{n}\), Appunti. In: Scuola Normale Superiore di Pisa (Nuova Serie) [Lecture Notes. Scuola Normale Superiore di Pisa (New Series)], 15, pp. 8–152. Edizioni della Normale, Pisa (2017)Google Scholar
  20. 20.
    Dipierro, S., Montoro, L., Peral, I., Sciunzi, B.: Qualitative properties of positive solutions to nonlocal critical problems involving the Hardy–Leray potential. Calc. Var. Partial Differ. Equ. 55(4), 99 (2016)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Dipierro, S., Valdinoci, E.: A density property for fractional weighted Sobolev spaces. Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 26(4), 397–422 (2015)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Fall, M.M.: Nonexistence of distributional supersolutions of a semilinear elliptic equation with Hardy potential. J. Funct. Anal. 264(3), 661–690 (2013)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Fall, M.M., Felli, V.: Unique continuation property and local asymptotics of solutions to fractional elliptic equations. Commun. Partial Differ. Equ. 39(2), 354–397 (2014)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Felmer, P., Quaas, A., Tan, J.: Positive solutions of the nonlinear Schrödinger equation with the fractional Laplacian. Proc. R. Soc. Edinb. Sect. A 142(6), 1237–1262 (2012)CrossRefGoogle Scholar
  25. 25.
    Fiscella, A., Pucci, P.: On certain nonlocal Hardy–Sobolev critical elliptic Dirichlet problems. Adv. Differ. Equ. 21(5–6), 571–599 (2016)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Fiscella, A., Pucci, P.: \(p\)-fractional Kirchhoff equations involving critical nonlinearities. Nonlinear Anal. Real World Appl. 35, 350–378 (2017)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Fiscella, A., Pucci, P., Saldi, S.: Existence of entire solutions for Schrödinger–Hardy systems involving two fractional operators. Nonlinear Anal. 158, 109–131 (2017)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Fiscella, A., Pucci, P., Zhang, B.: \(p\)-fractional Hardy-Schrödinger–Kirchhoff systems with critical nonlinearities. Adv. Nonlinear Anal.  https://doi.org/10.1515/anona-2018-0033
  29. 29.
    Frank, R.L., Lieb, E.H., Seiringer, R.: Hardy–Lieb–Thirring inequalities for fractional Schrödinger operators. J. Am. Math. Soc. 21(4), 925–950 (2008)CrossRefGoogle Scholar
  30. 30.
    Frank, R.L., Seiringer, R.: Non-linear ground state representations and sharp Hardy inequalities. J. Funct. Anal. 255(12), 3407–3430 (2008)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Frank, R.L.: On the uniqueness of ground states of non-local equations. J. Èqu Dériv. Partielles 5, 1–10 (2011)Google Scholar
  32. 32.
    Ghoussoub, N., Shakerian, S.: Borderline variational problems involving fractional Laplacians and critical singularities. Adv. Nonlinear Stud. 15(3), 527–555 (2015)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Kang, D., Kang, Y.: Quasilinear elliptic systems involving critical Hardy–Sobolev and Sobolev exponents. Bull. Malays. Math. Sci. Soc. 40(1), 1–17 (2017)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Kang, D., Peng, S.J.: Existence and asymptotic properties of solutions to elliptic systems involving multiple critical exponents. Sci. China Math. 54(2), 243–256 (2011)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Laskin, N.: Fractional quantum mechanics and Lévy path integrals. Phys. Lett. A 268(4–6), 298–305 (2000)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Lions, P.-L.: The concentration-compactness principle in the calculus of variations. The limit case I. Rev. Mat. Iberoamericana 1(1), 145–201 (1985)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Maz’ya, V., Shaposhnikova, T.: On the Bourgain, Brezis, and Mironescu theorem concerning limiting embeddings of fractional Sobolev spaces. J. Funct. Anal. 195(2), 230–238 (2002)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Mingqi, X., Molica Bisci, G., Tian, G., Zhang, B.: Infinitely many solutions for the stationary Kirchhoff problems involving the fractional \(p\)-Laplacian. Nonlinearity 29(2), 357–374 (2016)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Molica Bisci, G., Rădulescu, V.: Ground state solutions of scalar field fractional Schrödinger equations. Calc. Var. Partial Differ. Equ. 54(3), 2985–3008 (2015)CrossRefGoogle Scholar
  40. 40.
    Molica Bisci, G., Rădulescu, V., Servadei, R.: Variational methods for nonlocal fractional problems. In: With a foreword by Jean Mawhin. Encyclopedia of Mathematics and its Applications, vol. 162, pp. 97–383. Cambridge University Press, Cambridge (2016)Google Scholar
  41. 41.
    Molica Bisci, G., Repovs̆, D.: On doubly nonlocal fractional elliptic equations. Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl 26(2), 161–176 (2015)MathSciNetCrossRefGoogle Scholar
  42. 42.
    Palatucci, G., Pisante, A.: Improved Sobolev embeddings, profile decomposition, and concentration-compactness for fractional Sobolev spaces. Calc. Var. Partial Differ. Equ. 50(3–4), 799–829 (2014)MathSciNetCrossRefGoogle Scholar
  43. 43.
    Pucci, P., Xiang, M.Q., Zhang, B.L.: Multiple solutions for nonhomogeneous Schrödinger–Kirchhoff type equations involving the fractional \(p\)-Laplacian in \({\mathbb{R}}^{N}\). Calc. Var. Partial Differ. Equ. 54(3), 2785–2806 (2015)CrossRefGoogle Scholar
  44. 44.
    Rabinowitz, P.H.: Minimax methods in critical point theory with applications to differential equations, CBMS Regional Conference Series in Mathematics, 65. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI (1986). viii+100 ppGoogle Scholar
  45. 45.
    Rastegarzadeh, S., Nyamoradi, N.: Existence of Positive Solutions for Hardy nonlocal fractional elliptic equations involving critical nonlinearities, Topol. Meth. Nonlinear Anal. (2018) (Preperint)Google Scholar
  46. 46.
    Rüland, A.: Unique continuation for fractional Schrödinger equations with rough potentials. Commun. Partial Differ. Equ. 40(1), 77–114 (2015)CrossRefGoogle Scholar
  47. 47.
    Secchi, S.: Ground state solutions for nonlinear fractional Schrödinger equations in \({\mathbb{R}}^{N}\). J. Math. Phys. 54(3), 031501 (2013)MathSciNetCrossRefGoogle Scholar
  48. 48.
    Servadei, R., Valdinoci, E.: Mountain pass solutions for non-local elliptic operators. J. Math. Anal. Appl. 389(2), 887–898 (2012)MathSciNetCrossRefGoogle Scholar
  49. 49.
    Servadei, R., Valdinoci, E.: Variational methods for non-local operators of elliptic type. Discrete Contin. Dyn. Syst. 33(5), 2105–2137 (2013)MathSciNetzbMATHGoogle Scholar
  50. 50.
    Servadei, R., Valdinoci, E.: The Brezis–Nirenberg result for the fractional Laplacian. Trans. Am. Math. Soc. 367(1), 67–102 (2015)MathSciNetCrossRefGoogle Scholar
  51. 51.
    Wang, L., Zhang, B., Zhang, H.: Fractional Laplacian system involving doubly critical nonlinearities in \({\mathbb{R}}^{N}\). Electron. J. Qual. Theory Differ. Equ. 57, 17 (2017)zbMATHGoogle Scholar
  52. 52.
    Yang, J., Wu, F.: Doubly critical problems involving fractional Laplacians in \({\mathbb{R}}^{N}\). Adv. Nonlinear Stud. 17(4), 677–690 (2017)MathSciNetCrossRefGoogle Scholar

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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