Abstract
In this paper, we consider a nonlinear Schrödinger equation involving the fractional Laplacian with Dirichlet condition:
where \(\Omega \) is a domain (bounded or unbounded) in \(\mathbb R^n\) which is convex in \(x_1\)-direction. By using some ideas of maximum principle and the direct moving plane method, we prove that the solutions are strictly increasing in \(x_1\)-direction in the left half domain of \(\Omega \). Symmetry of some solutions are also proved. Meanwhile, we obtain a Liouville type theorem on the half space \(\mathbb R^n_+\).
Similar content being viewed by others
References
Abatangelo, N., Valdinoci, E.: Getting acquainted with the fractional laplacian (2017), preprint
Alves, C.O., Figueiredo, G.M., Pimenta, M.T.O.: Existence and profile of ground-state solutions to a 1-laplacian problem in R-N. Bullet. Brazil. Math. Soc. 51(3), 863–886 (2020)
Azzollini, A., D’Avenia, P., Pomponio, A.: On the SchrödingerCMaxwell equations under the effect of a general nonlinear term. Annal. De Linstitut Henri Poincare 27(2), 779–791 (2010)
Benhassine, A.: Fractional schrödinger equations with new conditions. Electron. J. Differ. Equ. 2018(5), 1–12 (2018)
Caffarelli, L., Silvestre, L.: Regularity theory for fully nonlinear integro-differential equations. Commun. Pure Appl. Math. 62(5), 597–638 (2009)
Cao, L., Wang, X., Dai, Z.: Radial Symmetry and Monotonicity of Solutions to a System Involving Fractional p-Laplacian in a Ball. Adv. Math. Phys. pp. Art. ID 1565,731, 6 (2018)
Chen, W.: Problems with more general operators and more general nonlinearities. Adv. Math. 308, 404–437 (2017)
Chen, W., Fang, Y., Yang, R.: Liouville theorems involving the fractional laplacian on a half space. Adv. Math. 274(2), 167–198 (2015)
Chen, W., Li, C.: Maximum principles for the fractional p-Laplacian and symmetry of solutions. Adv. Math. 335, 735–758 (2018)
Chen, W., Li, C., Li, Y.: A direct method of moving planes for the fractional laplacian. Adv. Math. 308, 404–437 (2017)
Chen, W., Li, Y., Ma, P.: The Fractional Laplacian. World Scientific Publishing Co., Singapore (2017)
Chen, W., Zhu, J.: Indefinite fractional elliptic problem and liouville theorems. J. Differ. Equ. 260(5), 4758–4785 (2016)
Cheng, T., Cheng, T., Cheng, T., Cheng, T., Cheng, T.: Monotonicity and symmetry of solutions to fractional Laplacian equation. Dis. Continuous Dyn. Syst. 37(7), 3587–3599 (2017)
Cheng, T., Huang, G., Li, C.: The maximum principles for fractional laplacian equations and their applications. Commun. Contemp. Math. (2016)
Felmer, P., Quaas, A., Tan, J.: Positive solutions of the nonlinear schrödinger equation with the fractional laplacian. Proc. Royal Soc. Edinburgh 142(6), 1237–1262 (2012)
Floer, A., Weinstein, A.: Nonspreading wave packets for the cubic schrödinger equation with a bounded potential. J. Funct. Anal. 69(3), 397–408 (1986)
Garofalo, N.: Fractional thoughts (2017), preprint
Ghoussoub, N., Robert, F.: The hardyCschrödinger operator with interior singularity: the remaining cases. Calculus Variat. Partial Differ. Equ. 56(5), 149 (2017)
Guliyev, V.S., Guliyev, R.V., Omarova, M.N., Ragusa, M.A.: Schrodinger type operators on local generalized Morrey spaces related to certain nonnegative potentials. Dis. Continuous Dyn. Syst. Ser. B 25(2), 671–690 (2020)
Joo, M.D., Ferraz, D.: Concentration-compactness at the mountain pass level for nonlocal schrödinger equations (2016)
Li, C., Wu, Z.: Acta Math. Sci. Ser. 38(5), 1567–1582 (2018)
Li, C., Wu, Z., Xu, H.: Bocher theorems and applications Proceedings. Natl. Acad. Sci. USA 115(27), 6976–6979 (2018)
Li, Y.: Remark on some conformally invariant integral equations: the method of moving spheres. J. Europ. Math. Soc. 2(2), 153–180 (2004)
Luo, H., Tang, X., Li, S.: Multiple solutions of nonlinear schrödinger equations with the fractional p-laplacian. Taiwanese J. Math. 21(5),(2017)
Ma, L., Chen, D.: Radial symmetry and monotonicity for an integral equation. J. Math. Anal. Appl. 342(2), 943–949 (2008)
Nezza, E.D., Palatucci, G., Valdinoci, E.: Hitchhikerös guide to the fractional sobolev spaces. Bullet. Des. Sci. Math. 136(5), 521–573 (2011)
Pino, M.D., Felmer, P.L.: Local mountain passes for semilinear elliptic problems in unbounded domains. Calculus of Variat. Partial Differ. Equ. 4(2), 121–137 (1996)
Rabinowitz, Paul H.: On a class of nonlinear schrödinger equations. Z. angew Math. Phys. 43, 270–291 (1992)
Secchi, S.: Ground state solutions for nonlinear fractional schrödinger equations in rn. J. Math. Phys. 54(3), 56108–305 (2013)
Secchi, S.: On fractional schrödinger equations in Rn without the ambrosetti-rabinowitz condition. Topol. Methods Nonlinear Anal. (2014)
Wang, Ying, Wang, J.: The method of moving planes for integral equation in an extremal case. J. Partial Differ. Equ. 3, 246–254 (2016)
Wang, X., Zeng, B.: On concentration of positive bound states of nonlinear schrödinger equations with competing potential functions. Commun. Math. Phys. 153(2), 229–244 (1993)
Acknowledgements
The authors would like to express their gratitude to referees for many valuable suggestions which help to improve the presentation of this paper and provide good directions for further research.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Maria Alessandra Ragusa.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
The first author was supported by the Science and Technology Research Program for the Education Department of Hubei province of China under Grant No. D20163101. The Second author was supported by the NNSF of China under Grant No. 11871096.
Rights and permissions
About this article
Cite this article
Yuan, L., Li, P. Symmetry and Monotonicity of a Nonlinear Schrödinger Equation Involving the Fractional Laplacian. Bull. Malays. Math. Sci. Soc. 44, 4109–4125 (2021). https://doi.org/10.1007/s40840-021-01158-z
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40840-021-01158-z