Abstract
We are concerned with the existence of least energy solutions of nonlinear Schrödinger equations involving the fractional Laplacian
for sufficiently large λ, 2 < p < \(\frac{{2N}} {{N - 2s}}\) for N ≥ 2. V (x) is a real continuous function on RN. Using variational methods we prove the existence of least energy solution u λ(x) which localizes near the potential well int V −1(0) for λ large. Moreover, if the zero sets int V −1(0) of V (x) include more than one isolated component, then u λ(x) will be trapped around all the isolated components. However, in Laplacian case s = 1, when the parameter λ is large, the corresponding least energy solution will be trapped around only one isolated component and become arbitrarily small in other components of int V −1(0). This is the essential difference with the Laplacian problems since the operator (−Δ)s is nonlocal.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Applebaum D. Lévy processes—from probability to finance and quantum groups. Notices Amer Math Soc, 2004, 51: 1336–1347
Barrios B, Colorado E, de Pablo A, et al. On some critical problems for the fractional Laplacian operator. J Differential Equations, 2012, 252: 6133–6162
Bona J L, Li Y A. Decay and analyticity of solitary waves. J Math Pures Appl, 1997, 76: 377–430
Cabré X, Sire Y. Nonlinear equations for fractional Laplacians I: Regularity, maximum principles, and Hamiltonian estimates. Ann Inst H Poincaré Anal Non Linéaire, 2014, 31: 23–53
Cabré X, Tan J. Positive solutions of nonlinear problems involving the square root of the Laplacian. Adv Math, 2010, 224: 2052–2093
Caffarelli L, Salsa S, Silvestre L. Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian. Invent Math, 2008, 171: 425–461
Caffarelli L, Silvestre L. An extension problem related to the fractional Laplacian. Comm Partial Differential Equations, 2007, 32: 1245–1260
Cheng M. Bound state for the fractional Schrödinger equation with unbounded potential. J Math Phys, 2012, 53: 043507
Choi W, Kim S, Lee K. Asymptotic behavior of solutions for nonlinear elliptic problems with the fractional Laplacian. J Funct Anal, 2014, 266: 6531–6598
Dávila J, del Pino M, Wei J. Concentrating standing waves for the fractional nonlinear Schrödinger equation. J Differential Equations, 2014, 256: 858–892
de Bouard A, Saut J C. Symmetries and decay of the generalized Kadomtsev-Petviashvili solitary waves. SIAM J Math Anal, 1997, 28: 1064–1085
Dipierro S, Palatucci G, Valdinoci E. Existence and symmetry results for a Schrödinger type problem involving the fractional Laplacian. Matematiche, 2013, 68: 201–216
Felmer P, Quaas A, Tan J. Positive solutions of the nonlinear Schrödinger equation with the fractional Laplacian. Proc Roy Soc Edinburgh Sect A, 2012, 142: 1237–1262
Frank R L, Lenzmann E. Uniqueness of non-linear ground states for fractional Laplacians in R. Acta Math, 2013, 210: 261–318
Jin T, Li Y, Xiong J. On a fractional Nirenberg problem, part I: Blow up analysis and compactness of solutions. J Eur Math Soc, 2014, 16: 1111–1171
Lions P L. The concentration-compactness principle in the calculus of variations: The locally compact case, part I. Ann Inst H Poincaré Anal Non linéaire, 1984, 1: 109–145
Maris M. On the existence, regularity and decay of solitary waves to a generalized Benjamin-Ono equation. Nonlinear Anal, 2002, 51: 1073–1085
Niu M, Tang Z. Least energy solutions of nonlinear Schrödinger equations involving the half Laplacian and potential wells. Commun Pure Appl Anal, 2016, 15: 1215–1231
Silvestre L. Regularity of the obstacle problem for a fractional power of the Laplace operator. Comm Pure Appl Math, 2007, 60: 67–112
Sire Y, Valdinoci E. Fractional Laplacian phase transitions and boundary reactions: A geometric inequality and a symmetry result. J Funct Anal, 2009, 256: 1842–1864
Tan J. The Brezis-Nirenberg type problem involving the square root of the Laplacian. Calc Var Partial Differential Equations, 2011, 42: 21–41
Tan J. Positive solutions for non local elliptic problems. Discrete Contin Dyn Syst, 2013, 33: 837–859
Tan J, Xiong J. A Harnack inequality for fractional Laplace equations with lower order terms. Discrete Contin Dyn Syst, 2011, 31: 975–983
Yan S, Yang J, Yu X. Equations involving fractional Laplacian operator: Compactness and application. J Funct Anal, 2015, 269: 47–79
Acknowledgements
This work was supported by National Natural Science Foundation of China (Grant No. 11171028).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Niu, M., Tang, Z. Least energy solutions of nonlinear Schrödinger equations involving the fractional Laplacian and potential wells. Sci. China Math. 60, 261–276 (2017). https://doi.org/10.1007/s11425-015-0830-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11425-015-0830-3