American Institute of Mathematical Sciences

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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

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 & 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.  Google Scholar

[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.  Google Scholar

[3]

A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381.  Google Scholar

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L. Bergé and A. Couairon, Nonlinear propagation of self-guided ultra-short pulses in ionized gases, Phys. Plasmas, 7 (2000), 210-230. 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), 207-219.  Google Scholar

[6]

F. Dalfovo et al., Theory of Bose-Einstein condensation in trapped gases, Rev. Mod. Phys., 71 (1999), 463-512. 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), 231-249. 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), 546-572.  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), 397-408. 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), 877-910.  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), 573-595.  Google Scholar

[12]

E. Lieb and M. Loss, "Analysis," Gradute Studies in Mathematics, AMS, Providence, Rhode island, 2001.  Google Scholar

[13]

P. L. Lions, The Choquard equation and related questions, Nonlinear Anal., 4 (1980), 1063-1072.  Google Scholar

[14]

A. G. Litvak, Self-focusing of powerful light beams by thermal effects, JETP Lett., 4 (1966), 230-232. Google Scholar

[15]

G. Menzala, On regular solutions of a nonlinear equation of Choquard's type, Proc. Roy. Soc. Edinburgh Sect. A, 86 (1980), 291-301.  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), 1499-1519.  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), 223-253. doi:  10.1007/BF02161413.  Google Scholar

[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.  Google Scholar

[19]

M. del Pino and P. Felmer, Semi-classical states of nonlinear Schrödinger equations: a variational reduction method, Math. Ann., 324 (2002), 1-32. doi: 10.1007/s002080200327.  Google Scholar

[20]

P. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Ang. Math. Phys., 43 (1992), 270-291. 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), 655-674.  Google Scholar

[22]

B. Sirakov, Standing wave solutions of the nonlinear Schrödinger equations in $R^N$, Annali di Matematica, 183 (2002), 73-83.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

N. Ackermann, On a periodic Schrödinger equation with nonlocal superlinear part, Math. Z., 248 (2004), 423-443.  Google Scholar

[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.  Google Scholar

[3]

A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381.  Google Scholar

[4]

L. Bergé and A. Couairon, Nonlinear propagation of self-guided ultra-short pulses in ionized gases, Phys. Plasmas, 7 (2000), 210-230. 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), 207-219.  Google Scholar

[6]

F. Dalfovo et al., Theory of Bose-Einstein condensation in trapped gases, Rev. Mod. Phys., 71 (1999), 463-512. 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), 231-249. 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), 546-572.  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), 397-408. 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), 877-910.  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), 573-595.  Google Scholar

[12]

E. Lieb and M. Loss, "Analysis," Gradute Studies in Mathematics, AMS, Providence, Rhode island, 2001.  Google Scholar

[13]

P. L. Lions, The Choquard equation and related questions, Nonlinear Anal., 4 (1980), 1063-1072.  Google Scholar

[14]

A. G. Litvak, Self-focusing of powerful light beams by thermal effects, JETP Lett., 4 (1966), 230-232. Google Scholar

[15]

G. Menzala, On regular solutions of a nonlinear equation of Choquard's type, Proc. Roy. Soc. Edinburgh Sect. A, 86 (1980), 291-301.  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), 1499-1519.  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), 223-253. doi:  10.1007/BF02161413.  Google Scholar

[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.  Google Scholar

[19]

M. del Pino and P. Felmer, Semi-classical states of nonlinear Schrödinger equations: a variational reduction method, Math. Ann., 324 (2002), 1-32. doi: 10.1007/s002080200327.  Google Scholar

[20]

P. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Ang. Math. Phys., 43 (1992), 270-291. 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), 655-674.  Google Scholar

[22]

B. Sirakov, Standing wave solutions of the nonlinear Schrödinger equations in $R^N$, Annali di Matematica, 183 (2002), 73-83.  Google Scholar

[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.  Google Scholar

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