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July  2017, 37(7): 3963-3987. doi: 10.3934/dcds.2017168

## Least energy solutions for nonlinear Schrödinger equation involving the fractional Laplacian and critical growth

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

Received  October 2015 Revised  March 2017 Published  April 2017

Fund Project: The second author was supported by National Science Foundation of China(11571040)

In this paper, we study a class of nonlinear Schrödinger equations involving the fractional Laplacian and the nonlinearity term with critical Sobolev exponent. We assume that the potential of the equations includes a parameter $λ$. Moreover, the potential behaves like a potential well when the parameter λ is large. Using variational methods, combining Nehari methods, we prove that the equation has a least energy solution which, as the parameter λ large, localizes near the bottom of the potential well. Moreover, if the zero set int $V^{-1}(0)$ of $V(x)$ includes more than one isolated component, then $u_\lambda (x)$ will be trapped around all the isolated components. However, in Laplacian case when $s=1$, for $\lambda$ large, the corresponding least energy solution will be trapped around only one isolated component and will become arbitrary 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.

Citation: Miaomiao Niu, Zhongwei Tang. Least energy solutions for nonlinear Schrödinger equation involving the fractional Laplacian and critical growth. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 3963-3987. doi: 10.3934/dcds.2017168
##### References:
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Anal., 266 (2014), 6531-6598. doi: 10.1016/j.jfa.2014.02.029. Google Scholar [16] S. Dipierro, G. Palatucci and E. Valdinoci, Existence and symmetry results for a Schrödinger type problem involving the fractional Laplacian, Matematiche (Catania), 68 (2013), 201-216. Google Scholar [17] P. Felmer, A. Quaas and J. Tan, Positive solutions of the nonlinear Schrödinger equation with the fractional Laplacian, Proc. Roy. Soc. Edinburgh., 142 (2012), 1237-1262. doi: 10.1017/S0308210511000746. Google Scholar [18] R. L. Frank and E. Lenzmann, Uniqueness of non-linear ground states for fractional Laplacians $\text{in} \Bbb R$, Acta Math., 210 (2013), 261-318. doi: 10.1007/s11511-013-0095-9. Google Scholar [19] R. L. Frank, E. Lenzmann and L. Silvestre, Uniqueness of radial solutions for the fractional Laplacian, Comm. Pure Appl. Math., 69 (2016), 1671-1726. doi: 10.1002/cpa.21591. Google Scholar [20] M. Gonzalez and J. Qing, Fractional conformal Laplacians and fractional Yamabe problems, Anal. PDE, 6 (2013), 1535-1576. doi: 10.2140/apde.2013.6.1535. Google Scholar [21] T. Jin, Y. Li and J. Xiong, On a fractional Nirenberg problem, part Ⅰ: Blow up analysis and compactness of solutions, J. Eur. Math. Soc., 16 (2014), 1111-1171. doi: 10.4171/JEMS/456. Google Scholar [22] M. Maris, On the existence, regularity and decay of solitary waves to a generalized Benjamin-Ono equation, Nonlin. Anal., 51 (2002), 1073-1085. doi: 10.1016/S0362-546X(01)00880-X. Google Scholar [23] M. Reed and B. Simon, Methods of Modern Mathematical Physics, Ⅳ. Analysis of Operators, Academic Press, New York, 1978. Google Scholar [24] L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math., 60 (2007), 67-112. doi: 10.1002/cpa.20153. Google Scholar [25] Y. Sire and E. Valdinoci, Fractional Laplacian phase transitions and boundary reactions: a geometric inequality and a symmetry result, J. Funct. Anal., 256 (2009), 1842-1864. doi: 10.1016/j.jfa.2009.01.020. Google Scholar [26] J. Tan, The Brezis-Nirenberg type problem involving the square root of the Laplacian, Calc. Var. Part. Diff. Equa., 42 (2011), 21-41. doi: 10.1007/s00526-010-0378-3. Google Scholar [27] J. Tan, Positive solutions for non local elliptic problems, Discrete Contin. Dyn. Syst., 33 (2013), 837-589. doi: 10.3934/dcds.2013.33.837. Google Scholar [28] J. Tan and J. Xiong, A Harnack inequality for fractional Laplace equations with lower order terms, Discrete Contin. Dyn. Syst., 31 (2011), 975-983. doi: 10.3934/dcds.2011.31.975. Google Scholar [29] Z. Tang, Least energy solutions for semilinear Schrödinger equations involving critical growth and indefinite potentials, Comm. Pure Appl. Anal., 13 (2014), 237-248. doi: 10.3934/cpaa.2014.13.237. Google Scholar [30] M. Weinstein, Solitary waves of nonlinear dispersive evolution equations with critical power nonlinearities, J. Diff. Equa., 69 (1987), 192-203. doi: 10.1016/0022-0396(87)90117-3. Google Scholar [31] M. Willem, Minimax theorems. Progr. Nonlinear Differential Equations and their Applications 24. Birkhäuser Boston, Inc. , Boston, MA, 1996. doi: 10.1007/978-1-4612-4146-1. Google Scholar [32] S. Yan, J. Yang and X. Yu, Equations involving fractional Laplacian operator: Compactness and application, J. Funct. Anal., 269 (2015), 47-79. doi: 10.1016/j.jfa.2015.04.012. Google Scholar

show all references

##### References:
 [1] A. Ambrosetti, M. Badiale and S. Cingolani, Semiclassical states of nonlinear Schrödinger equations, Arch. Ration. Mech. Anal., 140 (1997), 285-300. doi: 10.1007/s002050050067. Google Scholar [2] D. Applebaum, Lévy processes-from probability to finance and quantum groups, Notices Amer. Math. Soc., 51 (2004), 1336-1347. Google Scholar [3] B. Barrios, E. Colorado, A. de Pablo and U. Sánchez, On some critical problems for the fractional Laplacian operator, J. Diff. Equa., 252 (2012), 6133-6162. doi: 10.1016/j.jde.2012.02.023. Google Scholar [4] T. Bartsch, A. Pankov and Z. Wang, Nonlinear Schrödinger equations with steep pontential well, Comm. Contemp. Math., 3 (2001), 549-569. doi: 10.1142/S0219199701000494. Google Scholar [5] T. Bartsch and Z. Wang, Multiple positive solutions for a nonlinear Schrödinger equation, Z. Angew. Math. Phys., 51 (2000), 366-384. doi: 10.1007/PL00001511. Google Scholar [6] V. Benci and G. Cerami, Existence of positive solutions of the equation $-\triangle u+a(x)u=u^{\frac{N+2}{N-2}} \text{in}{\Bbb R}^{N}$, J. Funct. Anal., 88 (1990), 90-117. doi: 10.1016/0022-1236(90)90120-A. Google Scholar [7] J. L. Bona and Y. A. Li, Decay and analyticity of solitary waves, J. Math. Pures Appl., 76 (1997), 377-430. doi: 10.1016/S0021-7824(97)89957-6. Google Scholar [8] A. de Bouard and J. C. Saut, Symmetries and decay of the generalized Kadomtsev-Petviashvili solitary waves, SIAM J. Math. Anal., 28 (1997), 1064-1085. doi: 10.1137/S0036141096297662. Google Scholar [9] H. Brezis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), 486-490. doi: 10.2307/2044999. Google Scholar [10] X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians Ⅰ: Regularity, maximum principles, and Hamiltonian estimates, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 23-53. doi: 10.1016/j.anihpc.2013.02.001. Google Scholar [11] X. Cabré and J. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian, Adv. Math., 224 (2010), 2052-2093. doi: 10.1016/j.aim.2010.01.025. Google Scholar [12] L. Caffarelli, S. Salsa and L. Silvestre, Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian, Invent. Math., 171 (2008), 425-461. doi: 10.1007/s00222-007-0086-6. Google Scholar [13] L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. in Part. Diff. Equa., 32 (2007), 1245-1260. doi: 10.1080/03605300600987306. Google Scholar [14] M. Cheng, Bound state for the fractional Schrödinger equation with unbounded potential J. Math. Phys. 53 (2012), 043507, 7pp. doi: 10.1063/1.3701574. Google Scholar [15] W. Choi, S. Kim and K. Lee, Asymptotic behavior of solutions for nonlinear elliptic problems with the fractional Laplacian, J. Funct. Anal., 266 (2014), 6531-6598. doi: 10.1016/j.jfa.2014.02.029. Google Scholar [16] S. Dipierro, G. Palatucci and E. Valdinoci, Existence and symmetry results for a Schrödinger type problem involving the fractional Laplacian, Matematiche (Catania), 68 (2013), 201-216. Google Scholar [17] P. Felmer, A. Quaas and J. Tan, Positive solutions of the nonlinear Schrödinger equation with the fractional Laplacian, Proc. Roy. Soc. Edinburgh., 142 (2012), 1237-1262. doi: 10.1017/S0308210511000746. Google Scholar [18] R. L. Frank and E. Lenzmann, Uniqueness of non-linear ground states for fractional Laplacians $\text{in} \Bbb R$, Acta Math., 210 (2013), 261-318. doi: 10.1007/s11511-013-0095-9. Google Scholar [19] R. L. Frank, E. Lenzmann and L. Silvestre, Uniqueness of radial solutions for the fractional Laplacian, Comm. Pure Appl. Math., 69 (2016), 1671-1726. doi: 10.1002/cpa.21591. Google Scholar [20] M. Gonzalez and J. Qing, Fractional conformal Laplacians and fractional Yamabe problems, Anal. PDE, 6 (2013), 1535-1576. doi: 10.2140/apde.2013.6.1535. Google Scholar [21] T. Jin, Y. Li and J. Xiong, On a fractional Nirenberg problem, part Ⅰ: Blow up analysis and compactness of solutions, J. Eur. Math. Soc., 16 (2014), 1111-1171. doi: 10.4171/JEMS/456. Google Scholar [22] M. Maris, On the existence, regularity and decay of solitary waves to a generalized Benjamin-Ono equation, Nonlin. Anal., 51 (2002), 1073-1085. doi: 10.1016/S0362-546X(01)00880-X. Google Scholar [23] M. Reed and B. Simon, Methods of Modern Mathematical Physics, Ⅳ. Analysis of Operators, Academic Press, New York, 1978. Google Scholar [24] L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math., 60 (2007), 67-112. doi: 10.1002/cpa.20153. Google Scholar [25] Y. Sire and E. Valdinoci, Fractional Laplacian phase transitions and boundary reactions: a geometric inequality and a symmetry result, J. Funct. Anal., 256 (2009), 1842-1864. doi: 10.1016/j.jfa.2009.01.020. Google Scholar [26] J. Tan, The Brezis-Nirenberg type problem involving the square root of the Laplacian, Calc. Var. Part. Diff. Equa., 42 (2011), 21-41. doi: 10.1007/s00526-010-0378-3. Google Scholar [27] J. Tan, Positive solutions for non local elliptic problems, Discrete Contin. Dyn. Syst., 33 (2013), 837-589. doi: 10.3934/dcds.2013.33.837. Google Scholar [28] J. Tan and J. Xiong, A Harnack inequality for fractional Laplace equations with lower order terms, Discrete Contin. Dyn. Syst., 31 (2011), 975-983. doi: 10.3934/dcds.2011.31.975. Google Scholar [29] Z. Tang, Least energy solutions for semilinear Schrödinger equations involving critical growth and indefinite potentials, Comm. Pure Appl. Anal., 13 (2014), 237-248. doi: 10.3934/cpaa.2014.13.237. Google Scholar [30] M. Weinstein, Solitary waves of nonlinear dispersive evolution equations with critical power nonlinearities, J. Diff. Equa., 69 (1987), 192-203. doi: 10.1016/0022-0396(87)90117-3. Google Scholar [31] M. Willem, Minimax theorems. Progr. Nonlinear Differential Equations and their Applications 24. Birkhäuser Boston, Inc. , Boston, MA, 1996. doi: 10.1007/978-1-4612-4146-1. Google Scholar [32] S. Yan, J. Yang and X. Yu, Equations involving fractional Laplacian operator: Compactness and application, J. Funct. Anal., 269 (2015), 47-79. doi: 10.1016/j.jfa.2015.04.012. Google Scholar
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