# 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 and Applied Analysis, 2013, 12 (1) : 429-449. doi: 10.3934/cpaa.2013.12.429
##### References:
 [1] 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. [2] M. Colin and L. Jeanjean, Solutions for a quasilinear Schrödinge equation: a dual approach, Nonlinear Anal., 56 (2004), 213-226. doi: 10.1016/j.na.2003.09.008. [3] Y. H. Ding and A. Szulkin, Bound states for semilinear Schrödinger equations with sign-changing potential, Calc. Var. Partial Differential Equations, 29 (2007), 397-419. doi: 10.1007/s00526-006-0071-8. [4] 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. [5] Y. H. Ding and J. C. Wei, Semiclassical states for nonlinear Schrödinger equations with sign-changing potentials, J. Func. Anal., 251 (2007), 546-572. [6] M. del Pino and P. Felmer, Multipeak bound states of nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non Linéare, 15 (1998), 127-149. doi: 10.1016/S0294-1449(97)89296-7. [7] 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. [8] J. M. do Ó, O. Miyagaki and S. Soares, Soliton solutions for quasilinear Schrödinger equations: the critical exponential case, Nonlinear Anal., 67 (2007), 3357-3372. doi: 10.1016/j.na.2006.10.018. [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), 275-315. doi: 10.1007/s00526-009-0286-6. [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), 547-83. doi: 10.1142/S021919970900348X. [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), 1363-1386. [12] J. M. do Ó, O. Miyagaki and S. Soares, Soliton solutions for quasilinear Schrödinger equations with critical growth, J. Differential Equations, 248 (2010), 722-744. doi: 10.1016/j.jde.2009.11.030. [13] X. Kang and J. Wei, On interacting bumps of semi-classical states of nonlinear Schrödinger equations , Adv. Diff. Eqs., 5 (2000), 899-928. [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), 397-408. doi: 10.1016/0022-1236(86)90096-0. [15] P. L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case, Part II, Ann. Inst. H. Poincaré Anal. Non Linéare, 1 (1984), 223-283. [16] J. Liu and Z. Q. Wang, Soliton solutions for quasilinear Schrödinger equations I, Proc. Amer. Math. Soc. 131 (2002), 441-448. doi: 10.1090/S0002-9939-02-06783-7. [17] J. Liu, Y. Wang and Z. Q. Wang, Soliton solutions for quasilinear Schrödinger equations II, J. Differential Equations, 187 (2003), 473-493. doi: 10.1016/S0022-0396(02)00064-5. [18] J. Liu, Y. Wang and Z. Q. Wang, Solutions for quasilinear Schrödinger equations via the Nehari method, Comm. Partial Differential Equations, 29 (2004) 879-901. doi: 10.1081/PDE-120037335. [19] A. Moameni, Existence of soliton solutions for a quasilinear Schrödinger equation involving critical exponent in $R^N$, J. Differential Equations, 229 (2006), 570-587. doi: 10.1016/j.jde.2006.07.001. [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), 1499-1519. [21] 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. [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), 329-344. doi: 10.1007/s005260100105. [23] P. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Ang. Math. Phys., 43 (1992), 270-291. doi: 10.1007/BF00946631. [24] M. Reed and B. Simon, "Methods of Modern Mathematical Physics, IV Analysis of Operators," Academic Press, 1978. [25] B. Sirakov, Standing wave solutions of the nonlinear Schrödinger equations in $R^N$, Annali di Matematica, 183 (2002), 73-83. [26] E. B. Silva and G. F. Vieira, Quasilinear asymptotically periodic Schrödinger equations with critical growth, Calc. Var. Partial Differential Equations, 39 (2010), 1-33. doi: 10.1007/s00526-009-0299-1. [27] X. Wang, On concentration of positive bound states of nonlinear Schrödinger equations, Comm. Math. Phys., 153 (1993), 229-244. doi: 10.1007/BF02096642.

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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), 285-300. doi: 10.1007/s002050050067. [2] M. Colin and L. Jeanjean, Solutions for a quasilinear Schrödinge equation: a dual approach, Nonlinear Anal., 56 (2004), 213-226. doi: 10.1016/j.na.2003.09.008. [3] Y. H. Ding and A. Szulkin, Bound states for semilinear Schrödinger equations with sign-changing potential, Calc. Var. Partial Differential Equations, 29 (2007), 397-419. doi: 10.1007/s00526-006-0071-8. [4] 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. [5] Y. H. Ding and J. C. Wei, Semiclassical states for nonlinear Schrödinger equations with sign-changing potentials, J. Func. Anal., 251 (2007), 546-572. [6] M. del Pino and P. Felmer, Multipeak bound states of nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non Linéare, 15 (1998), 127-149. doi: 10.1016/S0294-1449(97)89296-7. [7] 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. [8] J. M. do Ó, O. Miyagaki and S. Soares, Soliton solutions for quasilinear Schrödinger equations: the critical exponential case, Nonlinear Anal., 67 (2007), 3357-3372. doi: 10.1016/j.na.2006.10.018. [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), 275-315. doi: 10.1007/s00526-009-0286-6. [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), 547-83. doi: 10.1142/S021919970900348X. [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), 1363-1386. [12] J. M. do Ó, O. Miyagaki and S. Soares, Soliton solutions for quasilinear Schrödinger equations with critical growth, J. Differential Equations, 248 (2010), 722-744. doi: 10.1016/j.jde.2009.11.030. [13] X. Kang and J. Wei, On interacting bumps of semi-classical states of nonlinear Schrödinger equations , Adv. Diff. Eqs., 5 (2000), 899-928. [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), 397-408. doi: 10.1016/0022-1236(86)90096-0. [15] P. L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case, Part II, Ann. Inst. H. Poincaré Anal. Non Linéare, 1 (1984), 223-283. [16] J. Liu and Z. Q. Wang, Soliton solutions for quasilinear Schrödinger equations I, Proc. Amer. Math. Soc. 131 (2002), 441-448. doi: 10.1090/S0002-9939-02-06783-7. [17] J. Liu, Y. Wang and Z. Q. Wang, Soliton solutions for quasilinear Schrödinger equations II, J. Differential Equations, 187 (2003), 473-493. doi: 10.1016/S0022-0396(02)00064-5. [18] J. Liu, Y. Wang and Z. Q. Wang, Solutions for quasilinear Schrödinger equations via the Nehari method, Comm. Partial Differential Equations, 29 (2004) 879-901. doi: 10.1081/PDE-120037335. [19] A. Moameni, Existence of soliton solutions for a quasilinear Schrödinger equation involving critical exponent in $R^N$, J. Differential Equations, 229 (2006), 570-587. doi: 10.1016/j.jde.2006.07.001. [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), 1499-1519. [21] 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. [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), 329-344. doi: 10.1007/s005260100105. [23] P. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Ang. Math. Phys., 43 (1992), 270-291. doi: 10.1007/BF00946631. [24] M. Reed and B. Simon, "Methods of Modern Mathematical Physics, IV Analysis of Operators," Academic Press, 1978. [25] B. Sirakov, Standing wave solutions of the nonlinear Schrödinger equations in $R^N$, Annali di Matematica, 183 (2002), 73-83. [26] E. B. Silva and G. F. Vieira, Quasilinear asymptotically periodic Schrödinger equations with critical growth, Calc. Var. Partial Differential Equations, 39 (2010), 1-33. doi: 10.1007/s00526-009-0299-1. [27] X. Wang, On concentration of positive bound states of nonlinear Schrödinger equations, Comm. Math. Phys., 153 (1993), 229-244. doi: 10.1007/BF02096642.
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