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August  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 and Continuous Dynamical Systems, 2017, 37 (7) : 3963-3987. doi: 10.3934/dcds.2017168
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. [2] D. Applebaum, Lévy processes-from probability to finance and quantum groups, Notices Amer. Math. Soc., 51 (2004), 1336-1347. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [23] M. Reed and B. Simon, Methods of Modern Mathematical Physics, Ⅳ. Analysis of Operators, Academic Press, New York, 1978. [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. [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. [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. [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. [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. [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. [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. [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. [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.

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. [2] D. Applebaum, Lévy processes-from probability to finance and quantum groups, Notices Amer. Math. Soc., 51 (2004), 1336-1347. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [23] M. Reed and B. Simon, Methods of Modern Mathematical Physics, Ⅳ. Analysis of Operators, Academic Press, New York, 1978. [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. [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. [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. [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. [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. [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. [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. [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. [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.
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