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Multi-peak Solutions of a Class of Fractional p-Laplacian Equations

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Abstract

This paper is devoted to a class of fractional p-Laplacian equations with the form

$$\begin{aligned} \varepsilon ^{ps}(-\Delta )_p^s u+V(x)|u|^{p-2}u=f(u)~~\text { in }~~{\mathbb {R}}^{N}, \end{aligned}$$

where \(s\in (0,1)\), \(N\ge 2\), \(1<p<+\infty \), \(N>ps\), \((-\Delta )_p^s\) stands for the fractional p-Laplacian. When f is superlinear and subcritical, we apply the variational gluing method and penalization technique to show the existence of families of positive multi-peak solutions concentrating around isolated components of local minimum of the potential V(x) as \(\varepsilon \rightarrow 0\).

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References

  1. Adams, R.A.: Sobolev Spaces. Academic Press, New York (1975)

    Google Scholar 

  2. Almgren, F., Lieb, E.: Symmetric decreasing rearrangement is sometimes continuous. J. Am. Math. Soc. 2, 683–773 (1989)

    MathSciNet  Google Scholar 

  3. Alves, C.O., Ambrosio, V., Isernia, T.: Existence, multiplicity and concentration for a class of fractional \(p\) &\(q\)-Laplacian problems in \({\mathbb{R} }^N\). Commun. Pure Appl. Anal. 18, 2009–2045 (2019)

    MathSciNet  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, Art. 47, p. 19 (2016)

  5. Ambrosio, V., Figueiredo, G.M., Isernia, T.: Existence and concentration of positive solutions for \(p\)-fractional Schrödinger equations. Ann. Math. Pura Appl. 199, 317–344 (2020)

    Google Scholar 

  6. Ambrosio, V., Isernia, T.: Multiplicity and concentration results for some nonlinear Schrödinger equations with the fractional \(p\)-Laplacian. Discrete Contin. Dyn. Syst. 38, 5835–5881 (2018)

    MathSciNet  Google Scholar 

  7. Ambrosio, V., Isernia, T.: A multiplicity result for a \((p, q)\)-Schrödinger-Kirchhoff type equation. Ann. Math. Pura Appl. 201, 943–984 (2022)

    Google Scholar 

  8. Ambrosio, V., Rădulescu, V.D.: Fractional double-phase patterns: concentration and multiplicity of solutions. J. Math. Pures Appl. 142, 101–145 (2020)

    MathSciNet  Google Scholar 

  9. An, X.M., Peng, S.J.: Multi-peak semiclassical bound states for fractional Schrödinger equations with fast decaying potentials. Electron. Res. Arch. 30, 585–614 (2022)

    MathSciNet  Google Scholar 

  10. Andreu, F., Mazón, J.M., Rossi, J.D., Toledo, J.: A nonlocal \(p\)-Laplacian evolution equation with nonhomogeneous Dirichlet boundary conditions. SIAM J. Math. Anal. 40, 1815-1851 (2008/2009)

  11. Bartolo, R., Molica Bisci, G.: Asymptotically linear fractional \(p\)-Laplacian equations. Ann. Math. Pura Appl. 196, 427–442 (2017)

    MathSciNet  Google Scholar 

  12. 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  Google Scholar 

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

    MathSciNet  Google Scholar 

  14. Bjorland, C., Caffarelli, L., Figalli, A.: Non-local gradient dependent operators. Adv. Math. 230, 1859–1894 (2012)

    MathSciNet  Google Scholar 

  15. Brasco, L., Mosconi, S., Squassina, M.: Optimal decay of extremal functions for the fractional Sobolev inequality. Calc. Var. Partial Differ. Equ. 55, 1–32 (2016)

    Google Scholar 

  16. Byeon, J., Jeanjean, L.: Standing waves for nonlinear Schrödinger equations with a general nonlinearity. Arch. Ration. Mech. Anal. 185, 185–200 (2007)

    MathSciNet  Google Scholar 

  17. Byeon, J., Jeanjean, L.: Multi-peak standing waves for nonlinear Schrödinger equations with a general nonlinearity. Discrete Contin. Dyn. Syst. 19, 255–269 (2007)

    MathSciNet  Google Scholar 

  18. Byeon, J., Jeanjean, L., Tanaka, K.: Standing waves for nonlinear Schrödinger equations with a general nonlinearity: one and two dimentional case. Commun. Partial Differ. Equ. 33, 1113–1136 (2008)

    Google Scholar 

  19. Byeon, J., Kwon, O., Seok, J.: Nonlinear scalar field equations involving the fractional Laplacian. Nonlinearity 30, 1659–1681 (2017)

    MathSciNet  Google Scholar 

  20. Byeon, J., Tanaka, K.: Semi-classical standing waves for nonlinear Schrödinger equations at structurally stable critical points of the potential. J. Eur. Math. Soc. 15, 1859–1899 (2013)

    MathSciNet  Google Scholar 

  21. Byeon, J., Wang, Z.-Q.: Standing waves with a critical frequency for nonlinear Schrödinger equations II. Calc. Var. Partial Differ. Equ. 18, 207–219 (2003)

    Google Scholar 

  22. Caffarelli, L.: Non-local Diffusions, Drifts and Games, Nonlinear Partial Differential Equations, pp. 37–52. Springer, Berlin (2012)

    Google Scholar 

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

    MathSciNet  Google Scholar 

  24. Chang, X.J., Niu, Z.H., Wang, Z.-Q.: Sign-changing solutions of fractional \(p\)-Laplacian problems. Adv. Nonlinear Stud. 1, 28–53 (2019)

    MathSciNet  Google Scholar 

  25. Chen, W.X., Li, C.M.: Maximum principles for the fractional \(p\)-Laplacian and symmetry of solutions. Adv. Math. 335, 735–758 (2018)

    MathSciNet  Google Scholar 

  26. Chen, W.X., Hu, Y.Y.: Monotonicity of positive solutions for nonlocal problems in unbounded domains. J. Funct. Anal. 281, Paper No. 109187, 32 (2021)

  27. Cingolani, S., Tanaka, K.: Semi-classical states for the nonlinear Choquard equtions: existence, multiplicity and concentration at a potential well. Rev. Mat. Iberoam. 35, 1885–1924 (2019)

    MathSciNet  Google Scholar 

  28. Correa, E., de Pablo, A.: Remarks on a nonlinear nonlocal operator in Orlicz spaces. Adv. Nonlinear Anal. 9, 305–326 (2020)

    MathSciNet  Google Scholar 

  29. Dai, W., Liu, Z., Wang, P. Y.: Monotonicity and symmetry of positive solutions to fractional p-Laplacian equation. Commun. Contemp. Math. 24, Paper No. 2150005, 17 (2022)

  30. Damascelli, L.: Comparison theorems for some quasilinear degenerate elliptic operators and applications to symmetry and monotonicity results. Ann. Inst. H. Poincáre Anal. Non Lineáire 15, 493–516 (1998)

  31. Dávila, J., del Pino, M., Wei, J.C.: Concentrating standing waves for the fractional nonlinear Schrödinger equation. J. Differ. Equ. 256, 858–892 (2014)

    Google Scholar 

  32. Del Pezzo, L.M., Quaas, A.: A Hopf’s lemma and a strong minimum principle for the fractional \(p\)-Laplacian. J. Differ. Equ. 263, 765–778 (2017)

    MathSciNet  Google Scholar 

  33. Del Pezzo, L.M., Quaas, A.: Spectrum of the fractional \(p\)-Laplacian in \({\mathbb{R}}^N\) and decay estimate for positive solutions of a Schrödinger equation. Nonlinear Anal. 193, 111479, 24 (2020)

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

    MathSciNet  Google Scholar 

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

    Google Scholar 

  36. Franzina, G., Palatucci, G.: Fractional \(p\)-eigenvalues. Riv. Math. Univ. Parma (N.S.) 5, 373–386 (2014)

    MathSciNet  Google Scholar 

  37. Gao, F.S., R\(\check{a}\)dulescu, V.D., Yang, M.B., Zheng, Y.: Standing waves for the pseudo-relativistic Hartree equation with Berestycki-Lions nonlinearity. J. Differ. Equ. 295, 70–112 (2021)

  38. Gloss, E.: Existence and concentration of bound states for a \(p\)-Laplacian equation in \({\mathbb{R} }^N\). Adv. Nonlinear Stud. 10, 273–296 (2010)

    MathSciNet  Google Scholar 

  39. He, X. M., Zou, W. M.: Existence and concentration result for the fractional Schrödinger equations with critical nonlinearities. Calc. Var. Partial Differ. Equ. 55, Art. 91, 39 (2016)

  40. He, Y., Luo, X., Rădulescu, V. D.: Nodal multi-peak standing waves of fourth-order Schrödinger equations with mixed dispersion. J. Geom. Anal. 32, Paper No. 30, 36 (2022)

  41. Iannizzotto, A.: Monotonicity of eigenvalues of the fractional \(p\)-Laplacian with singular weights. Topol. Methods Nonlinear Anal. 61, 423–443 (2023)

    MathSciNet  Google Scholar 

  42. Iannizzotto, A., Mosconi, S., Squassina, M.: Global Hölder regularity for the fractional \(p\)-Laplacian. Rev. Mat. Iberoam. 32, 1353–1392 (2016)

    MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

  45. Liu, Z., Wang, Z.-Q.: On the Ambrosetti-Rabinowitz superlinear condition. Adv. Nonlinear Stud. 4, 563–574 (2004)

    MathSciNet  Google Scholar 

  46. Mosconi, S., Perera, K., Squassina, M., Yang, Y.: The Brezis-Nirenberg problem for the fractional \(p\)-Laplacian. Calc. Var. Partial Differ. Equ. 55, 55–105 (2016)

    MathSciNet  Google Scholar 

  47. Perera, K., Squassina, M.: Bifurcation results for problems with fractional Trudinger-Moser nonlinearity. Discrete Contin. Dyn. Syst. Ser. S 11, 561–576 (2018)

    MathSciNet  Google Scholar 

  48. Piersanti, P., Pucci, P.: Entire solutions for critical \(p\)-fractional Hardy Schrödinger Kirchhoff equations. Publ. Mat. 62, 3–36 (2018)

    MathSciNet  Google Scholar 

  49. Rabinowitz, P.: Minimax Methods in Critical Point Theory with Applications to Differential Equations. American Mathematical Society, Providence, RI (1986)

    Google Scholar 

  50. Saito, T.: Existence of a positive solution for some quasilinear elliptic equations in \({\mathbb{R} }^N\). J. Differ. Equ. 338, 591–635 (2022)

    Google Scholar 

  51. Seok, J.: Spike-layer solutions to nonlinear fractional Schrödinger equations with almost optimal nonlinearities. Electron. J. Differ. Equ. 2015, 1–19 (2015)

    Google Scholar 

  52. Shang, X.D., Zhang, J.H.: Concentrating solutions of nonlinear fractional Schrödinger equation with potentials. J. Differ. Equ. 258, 1106–1128 (2015)

    Google Scholar 

  53. Shang, X.D., Zhang, J.H.: Multi-peak positive solutions for a fractional nonlinear elliptic equation. Discrete Contin. Dyn. Syst. 35, 3183–3201 (2015)

    MathSciNet  Google Scholar 

  54. Yang, J., Zhou, T.: Existence of single peak solutions for a nonlinear Schrödinger system with coupled quadratic nonlinearity. Adv. Nonlinear Anal. 11, 417–431 (2022)

    MathSciNet  Google Scholar 

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Acknowledgements

The authors express thanks to Prof. K. Tanaka and Prof. Z.-Q. Wang for some valuable discussion. This work was supported by JSPS KAKENHI(JP15K17567), NSFC(11971095, 12001044).

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Authors and Affiliations

  1. School of Mathematics and Statistics & Center for Mathematics and Interdisciplinary Sciences, Northeast Normal University, Changchun, 130024, Jilin, China

    Xiaojun Chang

  2. Department of Mathematics, Saitama University, Shimo-Okubo 255, Sakura-ku, Saitama-shi, 338-8570, Japan

    Yohei Sato

  3. Laboratory of Mathematics and Complex Systems (Ministry of Education), School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, China

    Chengxiang Zhang

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  1. Xiaojun Chang
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Chang, X., Sato, Y. & Zhang, C. Multi-peak Solutions of a Class of Fractional p-Laplacian Equations. J Geom Anal 34, 29 (2024). https://doi.org/10.1007/s12220-023-01475-8

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