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Existence and multiplicity of solutions for a class of fractional elliptic systems

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Abstract

In this paper we are concerned with the existence and multiplicity of solutions of the following nonlocal system involving the fractional Laplacian

$$\begin{aligned} (-\Delta )^\sigma u_i + a_i(x) u_i = f_i(x,u_1,\ldots ,u_m)\quad \text{ for }\;\; x \in \mathbb {R}^n \quad \text{ and }\;\; i=1,\ldots ,m, \end{aligned}$$

where \(\sigma \in (0,1)\), \(n \ge 1\), \((-\Delta )^\sigma \) denotes the fractional Laplacian of order \(\sigma \), \(a_i(x)\) are continuous and unbounded potentials which may change sign, and the nonlinearities \(f_i(x,u_1,\ldots ,u_m)\) are continuous functions which may be unbounded in x. We treat both the superquadratic situation and the nonquadratic situation at infinity on the nonlinearities \(f_i(x,u_1,\ldots ,u_m)\).

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References

  1. Ambrosetti, A., Rabinowitz, P.H.: Dual variational methods in critical point theory and applications. J. Funct. Anal. 14, 349–381 (1973)

    Article  MathSciNet  Google Scholar 

  2. Applebaum, D.: Lévy processes-From probability to finance quantum groups. Notices Am. Math. Soc. 51, 1336–1347 (2004)

    MATH  Google Scholar 

  3. Barrios, B., Colorado, E., de Pablo, A., Sánchez, U.: On some critical problems for the fractional Laplacian operator. J. Differ. Equ. 252, 6133–6162 (2012)

    Article  MathSciNet  Google Scholar 

  4. Bartolo, P., Benci, V., Fortunato, D.: Abstract critical point theorems and applications to some nonlinear problems with strong resonance at infinity. Nonlinear Anal. TMA 7, 981–1012 (1983)

    Article  MathSciNet  Google Scholar 

  5. Berestycki, H., Lions, P.-L.: Nonlinear scalar field equations I. Existence of a ground state. Arch. Ration. Mech. Anal. 82, 313–345 (1983)

    Article  MathSciNet  Google Scholar 

  6. Brandle, C., Colorado, E., de Pablo, A., Sánchez, U.: A concave-convex elliptic problem involving the fractional laplacian. Proc. R. Soc. Edinb. Sect. A. 143, 39–71 (2013)

    Article  MathSciNet  Google Scholar 

  7. Bucur, C., Valdinoci, E.: Nonlocal diffusion and applications. In: Lecture Notes of the Unione Matematica Italiana, 20. Springer, Unione Matematica Italiana, Bologna, (2016)

    Google Scholar 

  8. Brezis, H., Nirenberg, L.: Positive solutions of nonlinear elliptic equations involving critical Sobolev exponent. Commun. Pure Appl. Math. 36, 437–477 (1983)

    Article  MathSciNet  Google Scholar 

  9. Caffarelli, L.A.: Abel Symposium on Non-local Diffusions, Drifts and Games. Nonlinear Partial Differential Equations, vol. 7, pp. 37–52. Springer, Heidelberg (2012)

    MATH  Google Scholar 

  10. Caffarelli, L.A., Silvestre, L.: An extension problem related to the fractional Laplacian. Commun. Partial Differ. Equ. 32, 1245–1260 (2007)

    Article  MathSciNet  Google Scholar 

  11. Chang, X., Wang, Z.-Q.: Ground state of scalar field equations involving a fractional Laplacian with general nonlinearity. Nonlinearity 26, 479–494 (2013)

    Article  MathSciNet  Google Scholar 

  12. Cheng, M.: Bound state for the fractional Schrödinger equation with unbounded potential. J. Math. Phys. 53, 043507, 7 (2012)

    MATH  Google Scholar 

  13. Costa, D.G.: On a class of elliptic systems in \(\mathbb{R}^n\). Eletron. J. Differ. Eq. 7, 1–14 (1994)

    Google Scholar 

  14. Costa, D.G., Magalhães, C.A.: A variational approach to noncooperative elliptic systems. Nonlinear Anal. 25, 699–715 (1995)

    Article  MathSciNet  Google Scholar 

  15. de Souza, M., Araújo, Y.L.: Semilinear elliptic equations for the fractional Laplacian involving critical exponential growth. Math. Methods Appl. Sci. 40, 1757–1772 (2017)

    Article  MathSciNet  Google Scholar 

  16. de Souza, M., Araújo, Y.L.: On a class of fractional Schrödinger equations in \(\mathbb{R}^n\) with sign-changing potential. Applicable 04, 538–551 (2018)

    Article  Google Scholar 

  17. Dipierro, S., Pinamonti, A.: A geometric inequality and a symmetry result for elliptic systems involving the fractional Laplacian. J. Differ. Equ. 255, 85–119 (2013)

    Article  MathSciNet  Google Scholar 

  18. do Ó, J.M., Ferraz, D.: Concentration-compactness principle for nonlocal scalar field equations with critical growth. J. Math. Anal. Appl. 449, 1189–1228 (2017)

    Article  MathSciNet  Google Scholar 

  19. do Ó, J.M., Miyagaki, O.H., Squassina, M.: Critical and subcritical fractional problems with vainshing potentials. Commun. Contemp. Math. 18, 1550063 (2016)

    Article  MathSciNet  Google Scholar 

  20. Dipierro, S., Palatucci, G., Valdinoci, E.: Existence and symmetry results for a Schrödinger type problem involving the fractional Laplacian. Matematiche 68, 201–216 (2013)

    MathSciNet  MATH  Google Scholar 

  21. Felmer, P., Quaas, A., Tan, J.: Positive solutions of nonlinear Schrödinger equation with the fractional laplacian. Proc. R. Soc. Edinb. Sect. A Math. 142, 1237–1262 (2012)

    Article  Google Scholar 

  22. Fiscella, A., Pucci, P., Saldi, S.: Existence of entire solutions for Schrödinger–Hardy systems involving two fractional operators. Nonlinear Anal. 158, 109–131 (2017)

    Article  MathSciNet  Google Scholar 

  23. Gonçalves, J.V., Miyagaki, O.H.: Existence of nontrivial solutions for semilinear elliptic equations at resonance. Houst. J. Math. 16, 583–595 (1990)

    MathSciNet  MATH  Google Scholar 

  24. Guo, Y.: Nonexistence and symmetry of solutions to some fractional Laplacian equations in the upper half space. Acta Math. Sci. Ser. B Engl. Ed. 37, 836–851 (2017)

    Article  MathSciNet  Google Scholar 

  25. Guo, Z., Luo, S., Zou, W.: On critical systems involving fractional Laplacian. J. Math. Anal. Appl. 446, 681–706 (2017)

    Article  MathSciNet  Google Scholar 

  26. He, Q., Peng, S., Peng, Y.: Existence, non-degeneracy of proportional positive solutions and least energy solutions for a fractional elliptic system. Adv. Differ. Equ. 22, 867–892 (2017)

    MathSciNet  MATH  Google Scholar 

  27. He, X., Squassina, M., Zou, W.: The Nehari manifold for fractional systems involving critical nonlinearities. Commun. Pure Appl. Anal. 15, 1285–1308 (2016)

    Article  MathSciNet  Google Scholar 

  28. Laskin, N.: Fractional quantum mechanics and Lévy path integrals. Phys. Lett. A 268, 298–305 (2000)

    Article  MathSciNet  Google Scholar 

  29. Laskin, N.: Fractional Schrödinger equation. Phys. Rev. E 66, 056108 (2002)

    Article  MathSciNet  Google Scholar 

  30. Li, G., Zhou, H.: Multiple solutions to \(p-\)Laplacian problems with asymptotic nonlinearity as \(u^{p-1}\) at infinity. J. Lond. Math. Soc. 65, 123–138 (2002)

    Article  Google Scholar 

  31. Quaas, A., Xia, A.: A Liouville type theorem for Lane-Emden systems involving the fractional Laplacian. Nonlinearity 29, 2279–2297 (2016)

    Article  MathSciNet  Google Scholar 

  32. Rabinowitz, P.H.: Minimax methods in critical point theory with applications to differential equations. CBMS Regional Conf. Ser. in Math. 65, AMS, Providence, RI (1986)

  33. Rabinowitz, P.H.: On a class of nonlinear Schrödinger equations. Z. Angew Math. Phys. 43, 272–291 (1992)

    Article  Google Scholar 

  34. Servadei, R., Valdinoci, E.: Mountain Pass solutions for non-local elliptic operators. J. Math. Anal. Appl. 389, 887–898 (2012)

    Article  MathSciNet  Google Scholar 

  35. Servadei, R., Valdinoci, E.: Variational methods for non-local operators of elliptic type. Discrete Contin. Dyn. Syst. 33, 2105–2137 (2013)

    MathSciNet  MATH  Google Scholar 

  36. Servadei, R., Valdinoci, E.: Weak and viscosity solutions of the fractional Laplace equation. Publ. Math. 1, 133–154 (2014)

    Article  MathSciNet  Google Scholar 

  37. Sirakov, B.: Existence and multiplicity of solutions of semi-linear elliptic equations in \(\mathbb{R}^N\). Calc. Var. Partial Differ. Equ. 11, 119–142 (2000)

    Article  MathSciNet  Google Scholar 

  38. Shang, X., Zhang, J., Yang, Y.: On fractional Schödinger equation in \({\mathbb{R}}^N\) with critical growth. J. Math. Phys. 54, Article ID: 121502, 19 pp. (2013)

  39. Secchi, S.: Ground state solutions for nonlinear fractional Schrödinger equations in \(\mathbb{R}^n\). J. Math. Phys. 54, 031501 (2013)

    Article  MathSciNet  Google Scholar 

  40. Wang, Q.: Positive least energy solutions of fractional Laplacian systems with critical exponent. Electron. J. Differ. Equ. 116, (2016)

  41. Xiang, M., Zhang, B., Wei, Z.: Existence of solutions to a class of quasilinear Schrödinger system involving the fractional \(p-\)Laplacian. Electron. J. Qual. Theory Differ. Equ. 107, 1–15 (2016)

    Article  Google Scholar 

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Acknowledgements

The author would like to thank the referees for the careful review and the valuable comments, which provided insights that helped improve the paper. The author also would like to thank Princeton University, for its hospitality while part of this work was completed, and the Federal University of Paraíba for supporting his long term visit to Princeton University during the academic year 2017–2018.

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Correspondence to Manassés de Souza.

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This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001 and CNPq Grant 306498/2016-2.

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de Souza, M. Existence and multiplicity of solutions for a class of fractional elliptic systems. Collect. Math. 71, 103–122 (2020). https://doi.org/10.1007/s13348-019-00253-6

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