• Previous Article
    Dynamical properties of nonautonomous functional differential equations with state-dependent delay
  • DCDS Home
  • This Issue
  • Next Article
    Variational properties and linear stabilities of spatial isosceles orbits in the equal-mass three-body problem
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:
[1]

A. AmbrosettiM. 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. BarriosE. ColoradoA. 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. BartschA. 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. CaffarelliS. 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. ChoiS. 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. DipierroG. 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. FelmerA. 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. FrankE. 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. JinY. 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. YanJ. 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. AmbrosettiM. 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. BarriosE. ColoradoA. 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. BartschA. 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. CaffarelliS. 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. ChoiS. 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. DipierroG. 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. FelmerA. 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. FrankE. 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. JinY. 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. YanJ. 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

[1]

Zhongwei Tang. Least energy solutions for semilinear Schrödinger equations involving critical growth and indefinite potentials. Communications on Pure & Applied Analysis, 2014, 13 (1) : 237-248. doi: 10.3934/cpaa.2014.13.237

[2]

Miaomiao Niu, Zhongwei Tang. Least energy solutions of nonlinear Schrödinger equations involving the half Laplacian and potential wells. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1215-1231. doi: 10.3934/cpaa.2016.15.1215

[3]

Yinbin Deng, Wentao Huang. Least energy solutions for fractional Kirchhoff type equations involving critical growth. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 1929-1954. doi: 10.3934/dcdss.2019126

[4]

Jian Zhang, Shihui Zhu, Xiaoguang Li. Rate of $L^2$-concentration of the blow-up solution for critical nonlinear Schrödinger equation with potential. Mathematical Control & Related Fields, 2011, 1 (1) : 119-127. doi: 10.3934/mcrf.2011.1.119

[5]

Congming Peng, Dun Zhao. Global existence and blowup on the energy space for the inhomogeneous fractional nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3335-3356. doi: 10.3934/dcdsb.2018323

[6]

Pavel I. Naumkin, Isahi Sánchez-Suárez. On the critical nongauge invariant nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2011, 30 (3) : 807-834. doi: 10.3934/dcds.2011.30.807

[7]

Van Duong Dinh. On blow-up solutions to the focusing mass-critical nonlinear fractional Schrödinger equation. Communications on Pure & Applied Analysis, 2019, 18 (2) : 689-708. doi: 10.3934/cpaa.2019034

[8]

Zhengping Wang, Huan-Song Zhou. Radial sign-changing solution for fractional Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2016, 36 (1) : 499-508. doi: 10.3934/dcds.2016.36.499

[9]

Mohamad Darwich. On the $L^2$-critical nonlinear Schrödinger Equation with a nonlinear damping. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2377-2394. doi: 10.3934/cpaa.2014.13.2377

[10]

Jaeyoung Byeon, Sungwon Cho, Junsang Park. On the location of a peak point of a least energy solution for Hénon equation. Discrete & Continuous Dynamical Systems - A, 2011, 30 (4) : 1055-1081. doi: 10.3934/dcds.2011.30.1055

[11]

Wulong Liu, Guowei Dai. Multiple solutions for a fractional nonlinear Schrödinger equation with local potential. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2105-2123. doi: 10.3934/cpaa.2017104

[12]

Xudong Shang, Jihui Zhang. Multiplicity and concentration of positive solutions for fractional nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2239-2259. doi: 10.3934/cpaa.2018107

[13]

Patricio Felmer, César Torres. Radial symmetry of ground states for a regional fractional Nonlinear Schrödinger Equation. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2395-2406. doi: 10.3934/cpaa.2014.13.2395

[14]

Van Duong Dinh, Binhua Feng. On fractional nonlinear Schrödinger equation with combined power-type nonlinearities. Discrete & Continuous Dynamical Systems - A, 2019, 39 (8) : 4565-4612. doi: 10.3934/dcds.2019188

[15]

Vincenzo Ambrosio, Teresa Isernia. Multiplicity and concentration results for some nonlinear Schrödinger equations with the fractional p-Laplacian. Discrete & Continuous Dynamical Systems - A, 2018, 38 (11) : 5835-5881. doi: 10.3934/dcds.2018254

[16]

Vincenzo Ambrosio. Concentration phenomena for critical fractional Schrödinger systems. Communications on Pure & Applied Analysis, 2018, 17 (5) : 2085-2123. doi: 10.3934/cpaa.2018099

[17]

Kenji Nakanishi, Tristan Roy. Global dynamics above the ground state for the energy-critical Schrödinger equation with radial data. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2023-2058. doi: 10.3934/cpaa.2016026

[18]

Benoît Pausader. The focusing energy-critical fourth-order Schrödinger equation with radial data. Discrete & Continuous Dynamical Systems - A, 2009, 24 (4) : 1275-1292. doi: 10.3934/dcds.2009.24.1275

[19]

Yuxia Guo, Zhongwei Tang. Multi-bump solutions for Schrödinger equation involving critical growth and potential wells. Discrete & Continuous Dynamical Systems - A, 2015, 35 (8) : 3393-3415. doi: 10.3934/dcds.2015.35.3393

[20]

Lun Guo, Wentao Huang, Huifang Jia. Ground state solutions for the fractional Schrödinger-Poisson systems involving critical growth in $ \mathbb{R} ^{3} $. Communications on Pure & Applied Analysis, 2019, 18 (4) : 1663-1693. doi: 10.3934/cpaa.2019079

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (25)
  • HTML views (7)
  • Cited by (0)

Other articles
by authors

[Back to Top]