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Multiplicity and concentration of nontrivial nonnegative solutions for a fractional Choquard equation with critical exponent

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Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas Aims and scope Submit manuscript

Abstract

In present paper, we study the fractional Choquard equation

$$\begin{aligned} \varepsilon ^{2s}(-\Delta )^s u+V(x)u=\varepsilon ^{\mu -N}\left( \frac{1}{|x|^\mu }*F(u)\right) f(u)+|u|^{2^*_s-2}u \end{aligned}$$

where \(\varepsilon >0\) is a parameter, \(s\in (0,1),\)\(N>2s,\)\(2^*_s=\frac{2N}{N-2s}\) and \(0<\mu <\min \{2s,N-2s\}\). Under suitable assumption on V and f, we prove this problem has a nontrivial nonnegative ground state solution. Moreover, we relate the number of nontrivial nonnegative solutions with the topology of the set where the potential attains its minimum values and their’s concentration behavior.

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References

  1. Ackermann, N.: On a periodic Schrödinger equation with nonlocal superlinear part. Math. Z. 248(2), 423–443 (2004)

    MathSciNet  MATH  Google Scholar 

  2. Alves, C.O., Carrião, P.C., Medeiros, E.S.: Multiplicity of solutions for a class of quasilinear problem in exterior domains with Neumann conditions. Abstr. Appl. Anal. 3, 251–268 (2004)

    MathSciNet  MATH  Google Scholar 

  3. Alves, C.O., Gao, F., Squassina, M., Yang, M.: Singularly perturbed critical Choquard equations. J. Differ. Equ. 263(7), 3943–3988 (2017)

    MathSciNet  MATH  Google Scholar 

  4. Alves, C.O., Miyagaki, O.H.: Existence and concentration of solution for a class of fractional elliptic equation in \({\mathbb{R}}^N\) via penalization method. Calc. Var. Partial Differ. Equ. 55(3), 19 (2016)

    MATH  Google Scholar 

  5. Alves, C.O., Yang, M.: Investigating the multiplicity and concentration behaviour of solutions for a quasi-linear Choquard equation via the penalization method. Proc. R. Soc. Edinb. Sect. A 146(1), 23–58 (2016)

    MathSciNet  MATH  Google Scholar 

  6. Ambrosetti, A., Malchiodi, A.: Concentration phenomena for nonlinear Schrödinger equations: recent results and new perspectives. In Perspectives in Nonlinear Partial Differential Equations, Contemp. Math., Vol. 446, pp. 19–30. American Mathematical Society, Providence (2007)

  7. Ambrosio, V.: Multiplicity and concentration results for a fractional Choquard equation via penalization method. Potential Anal. 50(1), 55–82 (2017)

    MathSciNet  MATH  Google Scholar 

  8. Belchior, P., Bueno, H., Miyagaki, O.H., Pereira, G.A.: Remarks about a fractional Choquard equation: ground state, regularity and polynomial decay. Nonlinear Anal. 164, 38–53 (2017)

    MathSciNet  MATH  Google Scholar 

  9. Bhattarai, S.: On fractional Schrödinger systems of Choquard type. J. Differ. Equ. 263(6), 3197–3229 (2017)

    MathSciNet  MATH  Google Scholar 

  10. Brézis, H., Lieb, E.: A relation between pointwise convergence of functions and convergence of functionals. Proc. Am. Math. Soc. 88(3), 486–490 (1983)

    MathSciNet  MATH  Google Scholar 

  11. Bucur, C., Valdinoci, E.: Nonlocal Diffusion and Applications. Lecture Notes of the Unione Matematica Italiana, vol. 20. Springer, Berlin (2016)

    MATH  Google Scholar 

  12. Cassani, D., Zhang, J.: Choquard-type equations with Hardy–Littlewood–Sobolev upper-critical growth. Adv. Nonlinear Anal. 8(1), 1184–1212 (2019)

    MathSciNet  MATH  Google Scholar 

  13. Chen, Y., Liu, C.: Ground state solutions for non-autonomous fractional Choquard equations. Nonlinearity 29(6), 1827–1842 (2016)

    MathSciNet  MATH  Google Scholar 

  14. Cotsiolis, A., Tavoularis, N.: Best constants for Sobolev inequalities for higher order fractional derivatives. J. Math. Anal. Appl. 295(1), 225–236 (2004)

    MathSciNet  MATH  Google Scholar 

  15. d’Avenia, P., Siciliano, G., Squassina, M.: On fractional Choquard equations. Math. Models Methods Appl. Sci. 25(8), 1447–1476 (2015)

    MathSciNet  MATH  Google Scholar 

  16. del Pino, M., Felmer, P.L.: Local mountain passes for semilinear elliptic problems in unbounded domains. Calc. Var. Partial Differ. Equ. 4(2), 121–137 (1996)

    MathSciNet  MATH  Google Scholar 

  17. Di Nezza, E., Palatucci, G., Valdinoci, E.: Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136(5), 521–573 (2012)

    MathSciNet  MATH  Google Scholar 

  18. Gao, F., Yang, M.: The Brezis–Nirenberg type critical problem for the nonlinear Choquard equation. Sci. China Math. 61(7), 1219–1242 (2018)

    MathSciNet  MATH  Google Scholar 

  19. Gao, Z., Tang, X., Chen, S.: On existence and concentration behavior of positive ground state solutions for a class of fractional Schrödinger-Choquard equations. Z. Angew. Math. Phys. 69(5), 21 (2018)

    MATH  Google Scholar 

  20. Guo, L., Hu, T.: Existence and asymptotic behavior of the least energy solutions for fractional Choquard equations with potential well. Math. Methods Appl. Sci. 41(3), 1145–1161 (2018)

    MathSciNet  MATH  Google Scholar 

  21. He, X., Zou, W.: Existence and concentration result for the fractional Schrödinger equations with critical nonlinearities. Calc. Var. Partial Differ. Equ. 55(4), 39 (2016)

    MATH  Google Scholar 

  22. Laskin, N.: Fractional quantum mechanics and Lévy path integrals. Phys. Lett. A 268(4–6), 298–305 (2000)

    MathSciNet  MATH  Google Scholar 

  23. Laskin, N.: Fractional Schrödinger equation. Phys. Rev. E (3) 66(5), 056108 (2002)

    MathSciNet  Google Scholar 

  24. Lieb, E.H.: Existence and uniqueness of the minimizing solution of Choquard’s nonlinear equation. Stud. Appl. Math. 57(2), 93–105 (1976/1977)

    MathSciNet  MATH  Google Scholar 

  25. Lieb, E.H., Loss, M.: Analysis, 2nd edn., Graduate Studies in Mathematics, Vol. 14. American Mathematical Society, Providence (2001)

  26. Lions, P.-L.: The Choquard equation and related questions. Nonlinear Anal. 4(6), 1063–1072 (1980)

    MathSciNet  MATH  Google Scholar 

  27. Ma, L., Zhao, L.: Classification of positive solitary solutions of the nonlinear Choquard equation. Arch. Ration. Mech. Anal. 195(2), 455–467 (2010)

    MathSciNet  MATH  Google Scholar 

  28. Ma, P., Zhang, J.: Existence and multiplicity of solutions for fractional Choquard equations. Nonlinear Anal. 164, 100–117 (2017)

    MathSciNet  MATH  Google Scholar 

  29. Moroz, V., Van Schaftingen, J.: Groundstates of nonlinear Choquard equations: existence, qualitative properties and decay asymptotics. J. Funct. Anal. 265(2), 153–184 (2013)

    MathSciNet  MATH  Google Scholar 

  30. Moroz, V., Van Schaftingen, J.: Existence of groundstates for a class of nonlinear Choquard equations. Trans. Am. Math. Soc. 367(9), 6557–6579 (2015)

    MathSciNet  MATH  Google Scholar 

  31. Moroz, V., Van Schaftingen, J.: Semi-classical states for the Choquard equation. Calc. Var. Partial Differ. Equ. 52(1–2), 199–235 (2015)

    MathSciNet  MATH  Google Scholar 

  32. Moser, J.: A new proof of De Giorgi’s theorem concerning the regularity problem for elliptic differential equations. Commun. Pure Appl. Math. 13, 457–468 (1960)

    MathSciNet  MATH  Google Scholar 

  33. Mukheriee, T., Sreenadh, K.: Existence and multiplicity results for Brezis–Nirenberg type fractional Choquard equation. arXiv:1605.06805v1 [math.AP] (2016)

  34. Mukherjee, T., Sreenadh, K.: Fractional Choquard equation with critical nonlinearities. NoDEA Nonlinear Differ. Equ. Appl. 24(6), 34 (2017)

    MathSciNet  MATH  Google Scholar 

  35. 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)

    MathSciNet  MATH  Google Scholar 

  36. Pekar, S.: Untersuchung über die Elektronentheorie der Kristalle. Akademie Verlag, Berlin (1954)

    MATH  Google Scholar 

  37. Rabinowitz, P.H.: On a class of nonlinear Schrödinger equations. Z. Angew. Math. Phys. 43(2), 270–291 (1992)

    MathSciNet  MATH  Google Scholar 

  38. Secchi, S.: Ground state solutions for nonlinear fractional Schrödinger equations in \({\mathbb{R}}^N\). J. Math. Phys. 54(3), 17 (2013)

    MATH  Google Scholar 

  39. Shen, Z., Gao, F., Yang, M.: Ground states for nonlinear fractional Choquard equations with general nonlinearities. Math. Methods Appl. Sci. 39(14), 4082–4098 (2016)

    MathSciNet  MATH  Google Scholar 

  40. Szulkin, A., Weth, T.: The Method of Nehari Manifold. Handbook of Nonconvex Analysis and Applications, pp. 597–632. Int. Press, Somerville (2010)

    MATH  Google Scholar 

  41. Wang, F., Xiang, M.: Multiplicity of solutions to a nonlocal Choquard equation involving fractional magnetic operators and critical exponent. Electron. J. Differ. Equ. 306, 11 (2016)

    MathSciNet  MATH  Google Scholar 

  42. Wang, Y., Yang, Y.: Bifurcation results for the critical Choquard problem involving fractional p-Laplacian operator. Bound. Value Probl. 132, 11 (2018)

    MathSciNet  Google Scholar 

  43. Wei, J., Winter, M.: Strongly interacting bumps for the Schrödinger–Newton equations. J. Math. Phys. 50(1), 22 (2009)

    MATH  Google Scholar 

  44. Yang, Z.: Multiplicity and concentration behaviour of solutions for a fractional Choquard equation with critical growth (in press)

  45. Yang, Z., Zhao, F.: Three solutions for a fractional Schrödinger equation with vanishing potentials. Appl. Math. Lett. 76, 90–95 (2018)

    MathSciNet  MATH  Google Scholar 

  46. Zhang, H., Wang, J., Zhang, F.: Semiclassical states for fractional Choquard equations with critical growth. Commun. Pure Appl. Anal. 18(1), 519–538 (2019)

    MathSciNet  MATH  Google Scholar 

  47. Zhang, J., Zou, W.: Solutions concentrating around the saddle points of the potential for critical Schrödinger equations. Calc. Var. Partial Differ. Equ. 54(4), 4119–4142 (2015)

    MATH  Google Scholar 

  48. Zhang, W., Wu, X.: Nodal solutions for a fractional Choquard equation. J. Math. Anal. Appl. 464(2), 1167–1183 (2018)

    MathSciNet  MATH  Google Scholar 

  49. Zhong, C., Fan, X., Chen, W.: Introduction of Nonlinear Functional Analysis. Lanzhou University Publishing House, Lanzhou (1998)

    Google Scholar 

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Acknowledgements

We would like to thank the anonymous referee for his/her careful readings of our manuscript and the useful comments made for its improvement.

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  1. Department of Mathematics, Yunnan Normal University, Kunming, 650500, People’s Republic of China

    Shaoxiong Chen, Yue Li & Zhipeng Yang

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  1. Shaoxiong Chen
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Correspondence to Zhipeng Yang.

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This work is supported by NSFC (11771385, 11961081), China.

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Chen, S., Li, Y. & Yang, Z. Multiplicity and concentration of nontrivial nonnegative solutions for a fractional Choquard equation with critical exponent. RACSAM 114, 33 (2020). https://doi.org/10.1007/s13398-019-00768-4

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