March  2013, 12(2): 771-783. doi: 10.3934/cpaa.2013.12.771

Existence of solutions for singularly perturbed Schrödinger equations with nonlocal part

1. 

Department of Mathematics, Zhejiang Normal University, Jinhua, 321004

2. 

Institute of Mathematics, AMSS, Chinese Academy of Sciences, Beijing 100080

Received  July 2011 Revised  November 2011 Published  September 2012

In the present paper we study the existence of solutions for a nonlocal Schrödinger equation \begin{eqnarray*} -\varepsilon^2\Delta u +V(x)u =(\int_{R^3} \frac{|u|^{p}}{|x-y|^{\mu}}dy)|u|^{p-2}u, \end{eqnarray*} where $0 < \mu < 3$ and $\frac{6-\mu}{3} < p < {6-\mu}$. Under suitable assumptions on the potential $V(x)$, if the parameter $\varepsilon$ is small enough, we prove the existence of solutions by using Mountain-Pass Theorem.
Citation: Minbo Yang, Yanheng Ding. Existence of solutions for singularly perturbed Schrödinger equations with nonlocal part. Communications on Pure & Applied Analysis, 2013, 12 (2) : 771-783. doi: 10.3934/cpaa.2013.12.771
References:
[1]

N. Ackermann, On a periodic Schrödinger equation with nonlocal superlinear part,, Math. Z., 248 (2004), 423. Google Scholar

[2]

A. Ambrosetti, M. Badiale and S. Cingolani, Semiclassical states of nonlinear Schrödinger equations,, Arch. Rat. Mech. Anal., 140 (1997), 285. doi: 10.1007/s002050050067. Google Scholar

[3]

A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications,, J. Funct. Anal., 14 (1973), 349. Google Scholar

[4]

L. Bergé and A. Couairon, Nonlinear propagation of self-guided ultra-short pulses in ionized gases,, Phys. Plasmas, 7 (2000), 210. Google Scholar

[5]

J. Byeon and Z. Q. Wang, Standing waves with a critical frequency for nonlinear Schrödinger equations II,, Calc. Var. Partial Differential Equations, 18 (2003), 207. Google Scholar

[6]

F. Dalfovo et al., Theory of Bose-Einstein condensation in trapped gases,, Rev. Mod. Phys., 71 (1999), 463. Google Scholar

[7]

Y. H. Ding and F. H. Lin, Solutions of perturbed Schrödinger equations with critical nonlinearity,, Calc. Var. Partial Differential Equations, 30 (2007), 231. doi:  10.1007/s00526-007-0091-z. Google Scholar

[8]

Y. H. Ding and J. C. Wei, Semiclassical states for nonlinear Schrödinger equations with sign-changing potentials,, J. Func. Anal., 251 (2007), 546. Google Scholar

[9]

A. Floer and A. Weinstein, Nonspreading wave pachets for the packets for the cubic Schrödinger with a bounded potential,, J. Funct. Anal., 69 (1986), 397. doi: 10.1016/0022-1236(86)90096-0. Google Scholar

[10]

I. Ianni, Solutions of the Schrödinger-Poisson system concentrating on spheres, part II: existence,, Math. Models Methods Appl. Sci., 19 (2009), 877. Google Scholar

[11]

I. Ianni and G. Vaira, On concentration of positive bound states for the Schrödinger-Poisson problem with potentials,, Adv. Nonlin. Studies, 8 (2008), 573. Google Scholar

[12]

E. Lieb and M. Loss, "Analysis,", Gradute Studies in Mathematics, (2001). Google Scholar

[13]

P. L. Lions, The Choquard equation and related questions,, Nonlinear Anal., 4 (1980), 1063. Google Scholar

[14]

A. G. Litvak, Self-focusing of powerful light beams by thermal effects,, JETP Lett., 4 (1966), 230. Google Scholar

[15]

G. Menzala, On regular solutions of a nonlinear equation of Choquard's type,, Proc. Roy. Soc. Edinburgh Sect. A, 86 (1980), 291. Google Scholar

[16]

Y. G. Oh, Existence of semiclassical bound states of nonlinear Schrödinger equations with potentials of the class $(V)_\alpha$,, Comm. Part. Diff. Eqs., 13 (1988), 1499. Google Scholar

[17]

Y. G. Oh, On positive multi-lump bound states of nonlinear Schrödinger equations under multiple well potential,, Comm. Math. Phys., 131 (1990), 223. doi:  10.1007/BF02161413. Google Scholar

[18]

M. del Pino and P. Felmer, Multipeak bound states of nonlinear Schrödinger equations,, Ann. IHP, 15 (1998), 127. doi: 10.1016/S0294-1449(97)89296-7. Google Scholar

[19]

M. del Pino and P. Felmer, Semi-classical states of nonlinear Schrödinger equations: a variational reduction method,, Math. Ann., 324 (2002), 1. doi: 10.1007/s002080200327. Google Scholar

[20]

P. Rabinowitz, On a class of nonlinear Schrödinger equations,, Z. Ang. Math. Phys., 43 (1992), 270. doi: 10.1007/BF00946631. Google Scholar

[21]

D. Ruiz, The Schrödinger-Poisson equation under the effect of a nonlinear local term,, J. Funct. Anal., 237 (2006), 655. Google Scholar

[22]

B. Sirakov, Standing wave solutions of the nonlinear Schrödinger equations in $R^N$,, Annali di Matematica, 183 (2002), 73. Google Scholar

[23]

J. Wei and M. Winter, Strongly Interacting Bumps for the Schrodinger-Newton Equations,, Journal of Mathematical Physics, 50 (2009). doi: 10.1063/1.3060169. Google Scholar

show all references

References:
[1]

N. Ackermann, On a periodic Schrödinger equation with nonlocal superlinear part,, Math. Z., 248 (2004), 423. Google Scholar

[2]

A. Ambrosetti, M. Badiale and S. Cingolani, Semiclassical states of nonlinear Schrödinger equations,, Arch. Rat. Mech. Anal., 140 (1997), 285. doi: 10.1007/s002050050067. Google Scholar

[3]

A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications,, J. Funct. Anal., 14 (1973), 349. Google Scholar

[4]

L. Bergé and A. Couairon, Nonlinear propagation of self-guided ultra-short pulses in ionized gases,, Phys. Plasmas, 7 (2000), 210. Google Scholar

[5]

J. Byeon and Z. Q. Wang, Standing waves with a critical frequency for nonlinear Schrödinger equations II,, Calc. Var. Partial Differential Equations, 18 (2003), 207. Google Scholar

[6]

F. Dalfovo et al., Theory of Bose-Einstein condensation in trapped gases,, Rev. Mod. Phys., 71 (1999), 463. Google Scholar

[7]

Y. H. Ding and F. H. Lin, Solutions of perturbed Schrödinger equations with critical nonlinearity,, Calc. Var. Partial Differential Equations, 30 (2007), 231. doi:  10.1007/s00526-007-0091-z. Google Scholar

[8]

Y. H. Ding and J. C. Wei, Semiclassical states for nonlinear Schrödinger equations with sign-changing potentials,, J. Func. Anal., 251 (2007), 546. Google Scholar

[9]

A. Floer and A. Weinstein, Nonspreading wave pachets for the packets for the cubic Schrödinger with a bounded potential,, J. Funct. Anal., 69 (1986), 397. doi: 10.1016/0022-1236(86)90096-0. Google Scholar

[10]

I. Ianni, Solutions of the Schrödinger-Poisson system concentrating on spheres, part II: existence,, Math. Models Methods Appl. Sci., 19 (2009), 877. Google Scholar

[11]

I. Ianni and G. Vaira, On concentration of positive bound states for the Schrödinger-Poisson problem with potentials,, Adv. Nonlin. Studies, 8 (2008), 573. Google Scholar

[12]

E. Lieb and M. Loss, "Analysis,", Gradute Studies in Mathematics, (2001). Google Scholar

[13]

P. L. Lions, The Choquard equation and related questions,, Nonlinear Anal., 4 (1980), 1063. Google Scholar

[14]

A. G. Litvak, Self-focusing of powerful light beams by thermal effects,, JETP Lett., 4 (1966), 230. Google Scholar

[15]

G. Menzala, On regular solutions of a nonlinear equation of Choquard's type,, Proc. Roy. Soc. Edinburgh Sect. A, 86 (1980), 291. Google Scholar

[16]

Y. G. Oh, Existence of semiclassical bound states of nonlinear Schrödinger equations with potentials of the class $(V)_\alpha$,, Comm. Part. Diff. Eqs., 13 (1988), 1499. Google Scholar

[17]

Y. G. Oh, On positive multi-lump bound states of nonlinear Schrödinger equations under multiple well potential,, Comm. Math. Phys., 131 (1990), 223. doi:  10.1007/BF02161413. Google Scholar

[18]

M. del Pino and P. Felmer, Multipeak bound states of nonlinear Schrödinger equations,, Ann. IHP, 15 (1998), 127. doi: 10.1016/S0294-1449(97)89296-7. Google Scholar

[19]

M. del Pino and P. Felmer, Semi-classical states of nonlinear Schrödinger equations: a variational reduction method,, Math. Ann., 324 (2002), 1. doi: 10.1007/s002080200327. Google Scholar

[20]

P. Rabinowitz, On a class of nonlinear Schrödinger equations,, Z. Ang. Math. Phys., 43 (1992), 270. doi: 10.1007/BF00946631. Google Scholar

[21]

D. Ruiz, The Schrödinger-Poisson equation under the effect of a nonlinear local term,, J. Funct. Anal., 237 (2006), 655. Google Scholar

[22]

B. Sirakov, Standing wave solutions of the nonlinear Schrödinger equations in $R^N$,, Annali di Matematica, 183 (2002), 73. Google Scholar

[23]

J. Wei and M. Winter, Strongly Interacting Bumps for the Schrodinger-Newton Equations,, Journal of Mathematical Physics, 50 (2009). doi: 10.1063/1.3060169. Google Scholar

[1]

Christopher Grumiau, Marco Squassina, Christophe Troestler. On the Mountain-Pass algorithm for the quasi-linear Schrödinger equation. Discrete & Continuous Dynamical Systems - B, 2013, 18 (5) : 1345-1360. doi: 10.3934/dcdsb.2013.18.1345

[2]

Dorota Bors. Application of Mountain Pass Theorem to superlinear equations with fractional Laplacian controlled by distributed parameters and boundary data. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 29-43. doi: 10.3934/dcdsb.2018003

[3]

Woocheol Choi, Yong-Cheol Kim. The Malgrange-Ehrenpreis theorem for nonlocal Schrödinger operators with certain potentials. Communications on Pure & Applied Analysis, 2018, 17 (5) : 1993-2010. doi: 10.3934/cpaa.2018095

[4]

Ian Schindler, Kyril Tintarev. Mountain pass solutions to semilinear problems with critical nonlinearity. Conference Publications, 2007, 2007 (Special) : 912-919. doi: 10.3934/proc.2007.2007.912

[5]

Jianqing Chen. Sharp variational characterization and a Schrödinger equation with Hartree type nonlinearity. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 1613-1628. doi: 10.3934/dcdss.2016066

[6]

Jianqing Chen. A variational argument to finding global solutions of a quasilinear Schrödinger equation. Communications on Pure & Applied Analysis, 2008, 7 (1) : 83-88. doi: 10.3934/cpaa.2008.7.83

[7]

Yutian Lei. Liouville theorems and classification results for a nonlocal Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2018, 38 (11) : 5351-5377. doi: 10.3934/dcds.2018236

[8]

Dmitry Glotov, P. J. McKenna. Numerical mountain pass solutions of Ginzburg-Landau type equations. Communications on Pure & Applied Analysis, 2008, 7 (6) : 1345-1359. doi: 10.3934/cpaa.2008.7.1345

[9]

Claudianor O. Alves, Giovany M. Figueiredo, Marcelo F. Furtado. Multiplicity of solutions for elliptic systems via local Mountain Pass method. Communications on Pure & Applied Analysis, 2009, 8 (6) : 1745-1758. doi: 10.3934/cpaa.2009.8.1745

[10]

Hiroshi Isozaki, Hisashi Morioka. A Rellich type theorem for discrete Schrödinger operators. Inverse Problems & Imaging, 2014, 8 (2) : 475-489. doi: 10.3934/ipi.2014.8.475

[11]

Noboru Okazawa, Toshiyuki Suzuki, Tomomi Yokota. Energy methods for abstract nonlinear Schrödinger equations. Evolution Equations & Control Theory, 2012, 1 (2) : 337-354. doi: 10.3934/eect.2012.1.337

[12]

Mouhamed Moustapha Fall. Regularity estimates for nonlocal Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (3) : 1405-1456. doi: 10.3934/dcds.2019061

[13]

Claude Bardos, François Golse, Peter Markowich, Thierry Paul. On the classical limit of the Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2015, 35 (12) : 5689-5709. doi: 10.3934/dcds.2015.35.5689

[14]

Camille Laurent. Internal control of the Schrödinger equation. Mathematical Control & Related Fields, 2014, 4 (2) : 161-186. doi: 10.3934/mcrf.2014.4.161

[15]

D.G. deFigueiredo, Yanheng Ding. Solutions of a nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2002, 8 (3) : 563-584. doi: 10.3934/dcds.2002.8.563

[16]

Frank Wusterhausen. Schrödinger equation with noise on the boundary. Conference Publications, 2013, 2013 (special) : 791-796. doi: 10.3934/proc.2013.2013.791

[17]

Guan Huang. An averaging theorem for nonlinear Schrödinger equations with small nonlinearities. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3555-3574. doi: 10.3934/dcds.2014.34.3555

[18]

Shalva Amiranashvili, Raimondas  Čiegis, Mindaugas Radziunas. Numerical methods for a class of generalized nonlinear Schrödinger equations. Kinetic & Related Models, 2015, 8 (2) : 215-234. doi: 10.3934/krm.2015.8.215

[19]

Mingqi Xiang, Patrizia Pucci, Marco Squassina, Binlin Zhang. Nonlocal Schrödinger-Kirchhoff equations with external magnetic field. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1631-1649. doi: 10.3934/dcds.2017067

[20]

David Usero. Dark solitary waves in nonlocal nonlinear Schrödinger systems. Discrete & Continuous Dynamical Systems - S, 2011, 4 (5) : 1327-1340. doi: 10.3934/dcdss.2011.4.1327

2018 Impact Factor: 0.925

Metrics

  • PDF downloads (11)
  • HTML views (0)
  • Cited by (13)

Other articles
by authors

[Back to Top]