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 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.

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