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Existence of Solutions for Fractional-Choquard Equation with a Critical Exponential Growth in \({\mathbb {R}}^N\)

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Abstract

In this work, we study the existence of solutions to the fractional-Choquard equation

$$\begin{aligned} (-\Delta )^s_{\nu }u+V(x)|u|^{\nu -2}u=(I_{\alpha }*|u|^q)|u|^{q-2}u+h(u),\quad x\in {{\mathbb {R}}}^N, \end{aligned}$$
(0.1)

where \(\nu =\frac{N}{s}, 0<s<1, N\ge 2\), V(x) is a positive and bounded function in \({{\mathbb {R}}}^N\), \(I_{\alpha }\) is the Riesz potential, \(qs>N\) and the continuous function h(u) behaves like \(\exp (\alpha _0|u|^{\frac{N}{N-s}})\) growth. Using the symmetric rearrangement method with some special techniques and symmetric mountain pass lemma, we prove the existence of infinitely many solutions for (0.1) in \(W^{s,\nu }({\mathbb {R}}^N)\).

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References

  1. Adachi, S.J., Tanaka, K.: Trudinger type inequalities in \({\mathbb{R}}^N\) and their best exponents. Proc. Am. Math. Soc. 128(7), 2051–2057 (2000)

    Article  Google Scholar 

  2. Arora, R., Giacomoni, J., Mukherjee, T., Sreenadh, K.: n-Kirchhoff–Choquard equations with exponential nonlinearity. Nonlinear Anal. 186, 113–144 (2019)

    Article  MathSciNet  Google Scholar 

  3. Arora, R., Giacomoni, J., Mukherjee, T., Sreenadh, K.: Polyharmonic Kirchhoff type Choquard equations involving exponential nonlinearity with singular weights. Nonlinear Anal. 196, 111779 (2020)

    Article  MathSciNet  Google Scholar 

  4. Badiale, M., Serra, E.: Semilinear Elliptic Equations for Beginners, Existence Results via the Variational Approach. Springer, London (2011)

    Book  Google Scholar 

  5. Bartsch, T., Wang, Z.Q.: Existence and multiplicity results for some superlinear elliptic problems on \({\mathbb{R}}^N\). Commun. Partial Differ. Equ. 20, 1725–1741 (1995)

    Article  Google Scholar 

  6. 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)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  8. Bezerra doÓ, J.M.: \(N\)-Laplacian equations in \({\mathbb{R}}^{N}\) with critical growth. Abstr. Appl. Anal. 2, 301–315 (1997)

    Article  MathSciNet  Google Scholar 

  9. Bezerra doÓ, J.M., de Souza, M., de Medeiros, E., Severo, U.: An improvement for the Trudinger–Moser inequality and applications. J. Differ. Equ. 256, 1317–1349 (2014)

    Article  MathSciNet  Google Scholar 

  10. Bezerra doÓ, J.M., de Medeiros, E., Severo, U.: On a quasilinear nonhomogeneous elliptic equation with critical growth in \({\mathbb{R}}^N\). J. Differ. Equ. 246, 1363–1386 (2009)

    Article  Google Scholar 

  11. Bezerra doÓ, J.M., de Souza, M., de Medeiros, E., Severo, U.: Critical points for a functional involving critical growth of Trudinger–Moser type. Potential Anal. 42, 229–246 (2015)

    Article  MathSciNet  Google Scholar 

  12. Burchard, A.: A short course on rearrangement inequalities. http://www.doc88.com/p-1438088786 381.html

  13. Chen, C.S., Song, H.X.: Soliton solutions for quasilinear Schrödinger equation with critical exponential growth in \({\mathbb{R}}^N\). Appl. Math. 6(3), 317–337 (2016)

    Article  Google Scholar 

  14. Cingolani, S., Secchi, S.: Semiclassical analysis for pseudo-relativistic Hartree equations. J. Differ. Equ. 258, 4156–4179 (2015)

    Article  MathSciNet  Google Scholar 

  15. Cingolani, S., Secchi, S.: Ground states for the pseudo-relativistic Hartree equation with external potential. Proc. R. Soc. Edinb. A 145, 73–90 (2015)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  17. Edmunds, D.E., Ilyin, A.A.: Asymptotically sharp multiplicative inequalities. Bull. London Math. Soc. 27(1), 71–74 (1995)

    Article  MathSciNet  Google Scholar 

  18. Frank, R.L., Lenzmann, E.: On ground states for the \(L^2\)-critical boson star equation. arXiv:0910.2721 (2009)

  19. Giacomoni, J., Mishra, P.K., Sreenadh, K.: Fractional elliptic equations with critical exponential nonlinearity. Adv. Nonlinear Anal. 5(1), 57–74 (2016)

    Article  MathSciNet  Google Scholar 

  20. Giacomoni, J., Mishra, P.K., Sreenadh, K.: Critical growth problems for \(\frac{1}{2}\)-Laplacian in R. Differ. Equ. Appl. 8(3), 295–317 (2016)

    MathSciNet  MATH  Google Scholar 

  21. Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Springer, Berlin (1983)

    MATH  Google Scholar 

  22. Lenzmann, E.: Uniqueness of ground states for pseudorelativistic Hartree equations. Anal. PDE 2, 1–27 (2009)

    Article  MathSciNet  Google Scholar 

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

  24. Lieb, E.H., Loss, M.: Analysis, second ed. In: Graduate Studies in Mathematics, vol. 14, American Mathematical Society, Providence (2001)

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

    Article  MathSciNet  Google Scholar 

  26. Lions, P.L.: The concentration-compactness principle in the calculus of variations. The locally compact case I. Ann. Inst. H. Poincaré. Anal. Non Linéaire 1, 109–145 (1984)

    Article  MathSciNet  Google Scholar 

  27. Molica Bisci, G., Rădulescu, V., Servadei, R.: Variational Methods for Nonlocal Fractional Problems. Cambridge University Press, Cambridge (2016)

    Book  Google Scholar 

  28. Moroz, V., Van Schaftingen, J.: A guide to the Choquard equation. J. Fixed Point Theory Appl. 19, 773–813 (2017)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  30. Moser, J.: A sharp form of an inequality by N. Trudinger. Indiana Univ. Math. J. 20, 1077–1092 (1971)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  33. Pucci, P., Xiang, M.Q., Zhang, B.L.: Existence results for Schröinger–Choquard–Kirchhoff equations involving the fractional \(p\)-Laplacian. Adv. Calc. Var. (2017). https://doi.org/10.1515/acv-2016-0049

    Article  Google Scholar 

  34. Rabinowitz, P.H.: Minimax methods in critical point theory with application to differential equations. In: CBMS Reg. Conf. Ser. Math., vol. 65, Amer. Math. Soc., Providence (1986)

  35. Seok, J.: Nonlinear Choquard equations involving a critical local term. Appl. Math. Lett. 63, 77–87 (2017)

    Article  MathSciNet  Google Scholar 

  36. Souza, M.: On a singular elliptic problem involving critical growth in \({\mathbb{R}}^N\). Nonlinear Differ. Equ. Appl. 18, 199–215 (2011)

    Article  Google Scholar 

  37. Struwe, M.: Variational Methods. Springer, New York (2000)

    Book  Google Scholar 

  38. Trudinger, N.: On imbedding into Orlicz space and some applications. J. Math. Mech. 17, 473–484 (1967)

    MathSciNet  MATH  Google Scholar 

  39. Wu, D.: Existence and stability of standing waves for nonlinear fractional Schröinger equations with Hartree type nonlinearity. J. Math. Anal. Appl. 411(2), 530–542 (2014)

    Article  MathSciNet  Google Scholar 

  40. Xiang, M.Q., Zhang, B.L., Repovš, D.: Existence and multiplicity of solutions for fractional Schrödinger–Kirchhoff equations with Trudinger–Moser nonlinearity. Nonlinear Anal. 186, 74–98 (2019)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The author would like to express his sincere gratitude to the reviewers for the valuable comments and suggestions. Funding was provided by National Natural Science Foundation of China [Grant no. 11571092].

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  1. College of Science, Hohai University, Nanjing, 210098, China

    Caisheng Chen

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Chen, C. Existence of Solutions for Fractional-Choquard Equation with a Critical Exponential Growth in \({\mathbb {R}}^N\). Mediterr. J. Math. 17, 152 (2020). https://doi.org/10.1007/s00009-020-01592-6

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