Existence of solutions for singularly perturbed Schrödinger equations with nonlocal part

Previous Article
Multiplicity results for a class of elliptic problems with nonlinear boundary condition
CPAA Home
This Issue
Next Article
The Riemann problem of conservation laws in magnetogasdynamics

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

Existence of solutions for singularly perturbed Schrödinger equations with nonlocal part

Minbo Yang ^1, and Yanheng Ding ^2,

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

Abstract
Related Papers

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.

Keywords: Mountain-Pass Theorem, variational methods., Nonlocal Schrödinger equation.

Mathematics Subject Classification: Primary: 35J50, 35J60, 35J2.

Citation: Minbo Yang, Yanheng Ding. Existence of solutions for singularly perturbed Schrödinger equations with nonlocal part. Communications on Pure and 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), 423-443.
[2]	A. Ambrosetti, M. Badiale and S. Cingolani, Semiclassical states of nonlinear Schrödinger equations, Arch. Rat. Mech. Anal., 140 (1997), 285-300. doi: 10.1007/s002050050067.
[3]	A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381.
[4]	L. Bergé and A. Couairon, Nonlinear propagation of self-guided ultra-short pulses in ionized gases, Phys. Plasmas, 7 (2000), 210-230.
[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), 207-219.
[6]	F. Dalfovo et al., Theory of Bose-Einstein condensation in trapped gases, Rev. Mod. Phys., 71 (1999), 463-512.
[7]	Y. H. Ding and F. H. Lin, Solutions of perturbed Schrödinger equations with critical nonlinearity, Calc. Var. Partial Differential Equations, 30 (2007), 231-249. doi: 10.1007/s00526-007-0091-z.
[8]	Y. H. Ding and J. C. Wei, Semiclassical states for nonlinear Schrödinger equations with sign-changing potentials, J. Func. Anal., 251 (2007), 546-572.
[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), 397-408. doi: 10.1016/0022-1236(86)90096-0.
[10]	I. Ianni, Solutions of the Schrödinger-Poisson system concentrating on spheres, part II: existence, Math. Models Methods Appl. Sci., 19 (2009), 877-910.
[11]	I. Ianni and G. Vaira, On concentration of positive bound states for the Schrödinger-Poisson problem with potentials, Adv. Nonlin. Studies, 8 (2008), 573-595.
[12]	E. Lieb and M. Loss, "Analysis," Gradute Studies in Mathematics, AMS, Providence, Rhode island, 2001.
[13]	P. L. Lions, The Choquard equation and related questions, Nonlinear Anal., 4 (1980), 1063-1072.
[14]	A. G. Litvak, Self-focusing of powerful light beams by thermal effects, JETP Lett., 4 (1966), 230-232.
[15]	G. Menzala, On regular solutions of a nonlinear equation of Choquard's type, Proc. Roy. Soc. Edinburgh Sect. A, 86 (1980), 291-301.
[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), 1499-1519.
[17]	Y. G. Oh, On positive multi-lump bound states of nonlinear Schrödinger equations under multiple well potential, Comm. Math. Phys., 131 (1990), 223-253. doi: 10.1007/BF02161413.
[18]	M. del Pino and P. Felmer, Multipeak bound states of nonlinear Schrödinger equations, Ann. IHP, Analyse Nonlineaire, 15 (1998), 127-149. doi: 10.1016/S0294-1449(97)89296-7.
[19]	M. del Pino and P. Felmer, Semi-classical states of nonlinear Schrödinger equations: a variational reduction method, Math. Ann., 324 (2002), 1-32. doi: 10.1007/s002080200327.
[20]	P. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Ang. Math. Phys., 43 (1992), 270-291. doi: 10.1007/BF00946631.
[21]	D. Ruiz, The Schrödinger-Poisson equation under the effect of a nonlinear local term, J. Funct. Anal., 237 (2006), 655-674.
[22]	B. Sirakov, Standing wave solutions of the nonlinear Schrödinger equations in $R^N$, Annali di Matematica, 183 (2002), 73-83.
[23]	J. Wei and M. Winter, Strongly Interacting Bumps for the Schrodinger-Newton Equations, Journal of Mathematical Physics, 50 (2009), 012905. doi: 10.1063/1.3060169.

show all references

References:

[1]	N. Ackermann, On a periodic Schrödinger equation with nonlocal superlinear part, Math. Z., 248 (2004), 423-443.
[2]	A. Ambrosetti, M. Badiale and S. Cingolani, Semiclassical states of nonlinear Schrödinger equations, Arch. Rat. Mech. Anal., 140 (1997), 285-300. doi: 10.1007/s002050050067.
[3]	A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381.
[4]	L. Bergé and A. Couairon, Nonlinear propagation of self-guided ultra-short pulses in ionized gases, Phys. Plasmas, 7 (2000), 210-230.
[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), 207-219.
[6]	F. Dalfovo et al., Theory of Bose-Einstein condensation in trapped gases, Rev. Mod. Phys., 71 (1999), 463-512.
[7]	Y. H. Ding and F. H. Lin, Solutions of perturbed Schrödinger equations with critical nonlinearity, Calc. Var. Partial Differential Equations, 30 (2007), 231-249. doi: 10.1007/s00526-007-0091-z.
[8]	Y. H. Ding and J. C. Wei, Semiclassical states for nonlinear Schrödinger equations with sign-changing potentials, J. Func. Anal., 251 (2007), 546-572.
[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), 397-408. doi: 10.1016/0022-1236(86)90096-0.
[10]	I. Ianni, Solutions of the Schrödinger-Poisson system concentrating on spheres, part II: existence, Math. Models Methods Appl. Sci., 19 (2009), 877-910.
[11]	I. Ianni and G. Vaira, On concentration of positive bound states for the Schrödinger-Poisson problem with potentials, Adv. Nonlin. Studies, 8 (2008), 573-595.
[12]	E. Lieb and M. Loss, "Analysis," Gradute Studies in Mathematics, AMS, Providence, Rhode island, 2001.
[13]	P. L. Lions, The Choquard equation and related questions, Nonlinear Anal., 4 (1980), 1063-1072.
[14]	A. G. Litvak, Self-focusing of powerful light beams by thermal effects, JETP Lett., 4 (1966), 230-232.
[15]	G. Menzala, On regular solutions of a nonlinear equation of Choquard's type, Proc. Roy. Soc. Edinburgh Sect. A, 86 (1980), 291-301.
[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), 1499-1519.
[17]	Y. G. Oh, On positive multi-lump bound states of nonlinear Schrödinger equations under multiple well potential, Comm. Math. Phys., 131 (1990), 223-253. doi: 10.1007/BF02161413.
[18]	M. del Pino and P. Felmer, Multipeak bound states of nonlinear Schrödinger equations, Ann. IHP, Analyse Nonlineaire, 15 (1998), 127-149. doi: 10.1016/S0294-1449(97)89296-7.
[19]	M. del Pino and P. Felmer, Semi-classical states of nonlinear Schrödinger equations: a variational reduction method, Math. Ann., 324 (2002), 1-32. doi: 10.1007/s002080200327.
[20]	P. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Ang. Math. Phys., 43 (1992), 270-291. doi: 10.1007/BF00946631.
[21]	D. Ruiz, The Schrödinger-Poisson equation under the effect of a nonlinear local term, J. Funct. Anal., 237 (2006), 655-674.
[22]	B. Sirakov, Standing wave solutions of the nonlinear Schrödinger equations in $R^N$, Annali di Matematica, 183 (2002), 73-83.
[23]	J. Wei and M. Winter, Strongly Interacting Bumps for the Schrodinger-Newton Equations, Journal of Mathematical Physics, 50 (2009), 012905. doi: 10.1063/1.3060169.

[1]	Christopher Grumiau, Marco Squassina, Christophe Troestler. On the Mountain-Pass algorithm for the quasi-linear Schrödinger equation. Discrete and 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 and Continuous Dynamical Systems - B, 2018, 23 (1) : 29-43. doi: 10.3934/dcdsb.2018003
[3]	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
[4]	Woocheol Choi, Yong-Cheol Kim. The Malgrange-Ehrenpreis theorem for nonlocal Schrödinger operators with certain potentials. Communications on Pure and Applied Analysis, 2018, 17 (5) : 1993-2010. doi: 10.3934/cpaa.2018095
[5]	Hongwei Wang, Amin Esfahani. On the Cauchy problem for a nonlocal nonlinear Schrödinger equation. Discrete and Continuous Dynamical Systems - B, 2022 doi: 10.3934/dcdsb.2022039
[6]	Dmitry Glotov, P. J. McKenna. Numerical mountain pass solutions of Ginzburg-Landau type equations. Communications on Pure and Applied Analysis, 2008, 7 (6) : 1345-1359. doi: 10.3934/cpaa.2008.7.1345
[7]	Claudianor O. Alves, Giovany M. Figueiredo, Marcelo F. Furtado. Multiplicity of solutions for elliptic systems via local Mountain Pass method. Communications on Pure and Applied Analysis, 2009, 8 (6) : 1745-1758. doi: 10.3934/cpaa.2009.8.1745
[8]	Jianqing Chen. Sharp variational characterization and a Schrödinger equation with Hartree type nonlinearity. Discrete and Continuous Dynamical Systems - S, 2016, 9 (6) : 1613-1628. doi: 10.3934/dcdss.2016066
[9]	Jianqing Chen. A variational argument to finding global solutions of a quasilinear Schrödinger equation. Communications on Pure and Applied Analysis, 2008, 7 (1) : 83-88. doi: 10.3934/cpaa.2008.7.83
[10]	Yutian Lei. Liouville theorems and classification results for a nonlocal Schrödinger equation. Discrete and Continuous Dynamical Systems, 2018, 38 (11) : 5351-5377. doi: 10.3934/dcds.2018236
[11]	Hiroshi Isozaki, Hisashi Morioka. A Rellich type theorem for discrete Schrödinger operators. Inverse Problems and Imaging, 2014, 8 (2) : 475-489. doi: 10.3934/ipi.2014.8.475
[12]	Chenglin Wang, Jian Zhang. Cross-constrained variational method and nonlinear Schrödinger equation with partial confinement. Mathematical Control and Related Fields, 2021 doi: 10.3934/mcrf.2021036
[13]	Xiaobing Feng, Shu Ma. Stable numerical methods for a stochastic nonlinear Schrödinger equation with linear multiplicative noise. Discrete and Continuous Dynamical Systems - S, 2022, 15 (4) : 687-711. doi: 10.3934/dcdss.2021071
[14]	Noboru Okazawa, Toshiyuki Suzuki, Tomomi Yokota. Energy methods for abstract nonlinear Schrödinger equations. Evolution Equations and Control Theory, 2012, 1 (2) : 337-354. doi: 10.3934/eect.2012.1.337
[15]	Mouhamed Moustapha Fall. Regularity estimates for nonlocal Schrödinger equations. Discrete and Continuous Dynamical Systems, 2019, 39 (3) : 1405-1456. doi: 10.3934/dcds.2019061
[16]	Camille Laurent. Internal control of the Schrödinger equation. Mathematical Control and Related Fields, 2014, 4 (2) : 161-186. doi: 10.3934/mcrf.2014.4.161
[17]	Claude Bardos, François Golse, Peter Markowich, Thierry Paul. On the classical limit of the Schrödinger equation. Discrete and Continuous Dynamical Systems, 2015, 35 (12) : 5689-5709. doi: 10.3934/dcds.2015.35.5689
[18]	D.G. deFigueiredo, Yanheng Ding. Solutions of a nonlinear Schrödinger equation. Discrete and Continuous Dynamical Systems, 2002, 8 (3) : 563-584. doi: 10.3934/dcds.2002.8.563
[19]	Frank Wusterhausen. Schrödinger equation with noise on the boundary. Conference Publications, 2013, 2013 (special) : 791-796. doi: 10.3934/proc.2013.2013.791
[20]	Guan Huang. An averaging theorem for nonlinear Schrödinger equations with small nonlinearities. Discrete and Continuous Dynamical Systems, 2014, 34 (9) : 3555-3574. doi: 10.3934/dcds.2014.34.3555

2020 Impact Factor: 1.916

Tools

Metrics

Other articles
by authors

on AIMS
Minbo Yang Yanheng Ding
on Google Scholar
Minbo Yang Yanheng Ding

[Back to Top]