# American Institute of Mathematical Sciences

January  2013, 12(1): 429-449. doi: 10.3934/cpaa.2013.12.429

## Existence and multiplicity of semiclassical states for a quasilinear Schrödinger equation in $\mathbb{R}^N$

 1 Department of Mathematics, Zhejiang Normal University, Jinhua, 321004, China 2 Institute of Mathematics, AMSS, Chinese Academy of Sciences, Beijing 100080

Received  May 2011 Revised  March 2012 Published  September 2012

In this paper we consider the following modified version of nonlinear Schrödinger equation:

$-\varepsilon^2\Delta u +V(x)u-\varepsilon^2\Delta (u^2)u=g(x,u)$

in $\mathbb{R}^N$, $N\geq 3$ and $g(x,u)$ is a superlinear but subcritical function. Applying variational methods we show the existence and multiplicity of solutions provided $\varepsilon$ is sufficiently small enough.

Citation: Minbo Yang, Yanheng Ding. Existence and multiplicity of semiclassical states for a quasilinear Schrödinger equation in $\mathbb{R}^N$. Communications on Pure & Applied Analysis, 2013, 12 (1) : 429-449. doi: 10.3934/cpaa.2013.12.429
##### References:
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##### References:
 [1] 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 [2] M. Colin and L. Jeanjean, Solutions for a quasilinear Schrödinge equation: a dual approach,, Nonlinear Anal., 56 (2004), 213. doi: 10.1016/j.na.2003.09.008. Google Scholar [3] Y. H. Ding and A. Szulkin, Bound states for semilinear Schrödinger equations with sign-changing potential,, Calc. Var. Partial Differential Equations, 29 (2007), 397. doi: 10.1007/s00526-006-0071-8. Google Scholar [4] 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 [5] 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 [6] M. del Pino and P. Felmer, Multipeak bound states of nonlinear Schrödinger equations,, Ann. Inst. H. Poincar\'e Anal. Non Lin\'eare, 15 (1998), 127. doi: 10.1016/S0294-1449(97)89296-7. Google Scholar [7] 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 [8] J. M. do Ó, O. Miyagaki and S. Soares, Soliton solutions for quasilinear Schrödinger equations: the critical exponential case,, Nonlinear Anal., 67 (2007), 3357. doi: 10.1016/j.na.2006.10.018. Google Scholar [9] J. M. do Ó and U. Severo, Solitary waves for a class of quasilinear Schrödinger equations in dimension two,, Calc. Var. Partial Differential Equations, 38 (2010), 275. doi: 10.1007/s00526-009-0286-6. Google Scholar [10] J. M. do Ó, A. Moameni and U. Severo, Semi-classical states for quasilinear Schrödinger equations arising in plasma physics},, Commun. Contemp. Math., 11 (2009), 547. doi: 10.1142/S021919970900348X. Google Scholar [11] J. M. do Ó, E. Medeiros and U. Severo, On a quasilinear nonhomogeneous elliptic equation with critical growth in $R^N$,, J. Differential Equations, 246 (2009), 1363. Google Scholar [12] J. M. do Ó, O. Miyagaki and S. Soares, Soliton solutions for quasilinear Schrödinger equations with critical growth,, J. Differential Equations, 248 (2010), 722. doi: 10.1016/j.jde.2009.11.030. Google Scholar [13] X. Kang and J. Wei, On interacting bumps of semi-classical states of nonlinear Schrödinger equations ,, Adv. Diff. Eqs., 5 (2000), 899. Google Scholar [14] 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 [15] P. L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case, Part II,, Ann. Inst. H. Poincar\'e Anal. Non Lin\'eare, 1 (1984), 223. Google Scholar [16] J. Liu and Z. Q. Wang, Soliton solutions for quasilinear Schrödinger equations I,, Proc. Amer. Math. Soc. \textbf{131} (2002), 131 (2002), 441. doi: 10.1090/S0002-9939-02-06783-7. Google Scholar [17] J. Liu, Y. Wang and Z. Q. Wang, Soliton solutions for quasilinear Schrödinger equations II,, J. Differential Equations, 187 (2003), 473. doi: 10.1016/S0022-0396(02)00064-5. Google Scholar [18] J. Liu, Y. Wang and Z. Q. Wang, Solutions for quasilinear Schrödinger equations via the Nehari method,, Comm. Partial Differential Equations, 29 (2004), 879. doi: 10.1081/PDE-120037335. Google Scholar [19] A. Moameni, Existence of soliton solutions for a quasilinear Schrödinger equation involving critical exponent in $R^N$,, J. Differential Equations, 229 (2006), 570. doi: 10.1016/j.jde.2006.07.001. Google Scholar [20] 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 [21] 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 [22] M. Poppenberg, K. Schmitt and Z. Q. Wang, On the existence of soliton solutions to quasilinear Schrödinger equations,, Calc. Var. Partial Differential Equations, 14 (2002), 329. doi: 10.1007/s005260100105. Google Scholar [23] P. Rabinowitz, On a class of nonlinear Schrödinger equations,, Z. Ang. Math. Phys., 43 (1992), 270. doi: 10.1007/BF00946631. Google Scholar [24] M. Reed and B. Simon, "Methods of Modern Mathematical Physics, IV Analysis of Operators,", Academic Press, (1978). Google Scholar [25] B. Sirakov, Standing wave solutions of the nonlinear Schrödinger equations in $R^N$,, Annali di Matematica, 183 (2002), 73. Google Scholar [26] E. B. Silva and G. F. Vieira, Quasilinear asymptotically periodic Schrödinger equations with critical growth,, Calc. Var. Partial Differential Equations, 39 (2010), 1. doi: 10.1007/s00526-009-0299-1. Google Scholar [27] X. Wang, On concentration of positive bound states of nonlinear Schrödinger equations,, Comm. Math. Phys., 153 (1993), 229. doi: 10.1007/BF02096642. Google Scholar
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