# American Institute of Mathematical Sciences

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:
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##### 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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