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:
[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

show all references

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

[1]

Christopher Grumiau, Marco Squassina, Christophe Troestler. On the Mountain-Pass algorithm for the quasi-linear Schrödinger equation. Discrete & Continuous Dynamical Systems - B, 2013, 18 (5) : 1345-1360. doi: 10.3934/dcdsb.2013.18.1345

[2]

Silvia Cingolani, Mónica Clapp. Symmetric semiclassical states to a magnetic nonlinear Schrödinger equation via equivariant Morse theory. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1263-1281. doi: 10.3934/cpaa.2010.9.1263

[3]

Harald Friedrich. Semiclassical and large quantum number limits of the Schrödinger equation. Conference Publications, 2003, 2003 (Special) : 288-294. doi: 10.3934/proc.2003.2003.288

[4]

Chang-Lin Xiang. Remarks on nondegeneracy of ground states for quasilinear Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 2016, 36 (10) : 5789-5800. doi: 10.3934/dcds.2016054

[5]

Dorota Bors. Application of Mountain Pass Theorem to superlinear equations with fractional Laplacian controlled by distributed parameters and boundary data. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 29-43. doi: 10.3934/dcdsb.2018003

[6]

Ian Schindler, Kyril Tintarev. Mountain pass solutions to semilinear problems with critical nonlinearity. Conference Publications, 2007, 2007 (Special) : 912-919. doi: 10.3934/proc.2007.2007.912

[7]

Patricio Felmer, César Torres. Radial symmetry of ground states for a regional fractional Nonlinear Schrödinger Equation. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2395-2406. doi: 10.3934/cpaa.2014.13.2395

[8]

Alex H. Ardila. Stability of ground states for logarithmic Schrödinger equation with a $δ^{\prime}$-interaction. Evolution Equations & Control Theory, 2017, 6 (2) : 155-175. doi: 10.3934/eect.2017009

[9]

Wentao Huang, Jianlin Xiang. Soliton solutions for a quasilinear Schrödinger equation with critical exponent. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1309-1333. doi: 10.3934/cpaa.2016.15.1309

[10]

Kun Cheng, Yinbin Deng. Nodal solutions for a generalized quasilinear Schrödinger equation with critical exponents. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 77-103. doi: 10.3934/dcds.2017004

[11]

Jianqing Chen. A variational argument to finding global solutions of a quasilinear Schrödinger equation. Communications on Pure & Applied Analysis, 2008, 7 (1) : 83-88. doi: 10.3934/cpaa.2008.7.83

[12]

Xiang-Dong Fang. A positive solution for an asymptotically cubic quasilinear Schrödinger equation. Communications on Pure & Applied Analysis, 2019, 18 (1) : 51-64. doi: 10.3934/cpaa.2019004

[13]

Silvia Cingolani, Mnica Clapp, Simone Secchi. Intertwining semiclassical solutions to a Schrödinger-Newton system. Discrete & Continuous Dynamical Systems - S, 2013, 6 (4) : 891-908. doi: 10.3934/dcdss.2013.6.891

[14]

Rémi Carles, Christof Sparber. Semiclassical wave packet dynamics in Schrödinger equations with periodic potentials. Discrete & Continuous Dynamical Systems - B, 2012, 17 (3) : 759-774. doi: 10.3934/dcdsb.2012.17.759

[15]

Dmitry Glotov, P. J. McKenna. Numerical mountain pass solutions of Ginzburg-Landau type equations. Communications on Pure & Applied Analysis, 2008, 7 (6) : 1345-1359. doi: 10.3934/cpaa.2008.7.1345

[16]

Claudianor O. Alves, Giovany M. Figueiredo, Marcelo F. Furtado. Multiplicity of solutions for elliptic systems via local Mountain Pass method. Communications on Pure & Applied Analysis, 2009, 8 (6) : 1745-1758. doi: 10.3934/cpaa.2009.8.1745

[17]

Yinbin Deng, Yi Li, Xiujuan Yan. Nodal solutions for a quasilinear Schrödinger equation with critical nonlinearity and non-square diffusion. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2487-2508. doi: 10.3934/cpaa.2015.14.2487

[18]

Yaotian Shen, Youjun Wang. A class of generalized quasilinear Schrödinger equations. Communications on Pure & Applied Analysis, 2016, 15 (3) : 853-870. doi: 10.3934/cpaa.2016.15.853

[19]

GUANGBING LI. Positive solution for quasilinear Schrödinger equations with a parameter. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1803-1816. doi: 10.3934/cpaa.2015.14.1803

[20]

Hiroshi Isozaki, Hisashi Morioka. A Rellich type theorem for discrete Schrödinger operators. Inverse Problems & Imaging, 2014, 8 (2) : 475-489. doi: 10.3934/ipi.2014.8.475

2018 Impact Factor: 0.925

Metrics

  • PDF downloads (11)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]