March  2012, 11(2): 829-844. doi: 10.3934/cpaa.2012.11.829

Multi-bump solutions for a class of quasilinear equations on $R$

1. 

Universidade Federal da Campina Grande, Departamento de Matemática, 58109-970, Campina Grande - PB

2. 

Departmento de Matemática, Universidade Federal de Juiz de Fora, 36036-330, Juiz de Fora, MG, Brazil

3. 

Departmento de Matemática, Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, 13560-970, São Carlos, SP, Brazil

Received  January 2010 Revised  July 2011 Published  October 2011

This paper is concerned with the existence of multi-bump solutions to a class of quasilinear Schrödinger equations in $R$. The proof relies on variational methods and combines some arguments given by del Pino and Felmer, Ding and Tanaka, and Séré.
Citation: Claudianor O. Alves, Olímpio H. Miyagaki, Sérgio H. M. Soares. Multi-bump solutions for a class of quasilinear equations on $R$. Communications on Pure & Applied Analysis, 2012, 11 (2) : 829-844. doi: 10.3934/cpaa.2012.11.829
References:
[1]

C. O. Alves, Existence of multi-bump solutions for a class of quasilinear problems,, Adv. Nonlinear Stud., 6 (2006), 491. Google Scholar

[2]

C. O. Alves, D. C. de Morais Filho and M. A. S. Souto, Multiplicity of positive solutions for a class of problems with critical growth in $\mathbbR^N$,, Proc. Edinburgh Math. Soc., 52 (2009), 1. doi: 10.1017/S0013091507000028. Google Scholar

[3]

C. O. Alves and M. A. S. Souto, Multiplicity of positive solutions for a class of problems with exponential critical growth in $R^2$,, J. Differential Equations, 244 (2008), 1502. doi: 10.1016/j.jde.2007.09.007. Google Scholar

[4]

M. J. Alves, P. C. Carrião and O. H. Miyagaki, Soliton solutions to a class of quasilinear elliptic equations on $\mathbbR$,, Adv. Nonlinear Stud., 7 (2007), 579. Google Scholar

[5]

A. Ambrosetti and Z. Q. Wang, Positive solutions to a class of quasilinear elliptic equation on $\mathbbR$,, Discrete Contin. Dyn. Syst., 9 (2003), 55. doi: 10.3934/dcds.2003.9.55. Google Scholar

[6]

T. Bartsch and Z. Q. Wang, Existence and multiplicity results for some superlinear elliptic problems on $\mathbbR^N$,, Comm. Part. Diff. Eqs., 20 (1995), 1725. doi: 10.1080/03605309508821149. Google Scholar

[7]

T. Bartsch and Z. Q. Wang, Multiple positive solutions for a nonlinear Schrodinger equation,, Z. Angew Math. Phys., 51 (2000), 366. doi: 10.1007/PL00001511. Google Scholar

[8]

T. Bartsch, A. Pankov and Z. Q. Wang, Nonlinear Schrödinger equations with steep potential well,, Comm. Contemp. Math., 3 (2001), 549. doi: 10.1142/S0219199701000494. Google Scholar

[9]

J. Chen and B. Guo, Multiple nodal bound states for a quasilinear Schrödinger equations,, J. Math. Phys., 46 (2005). doi: 10.1063/1.2138045. Google Scholar

[10]

M. Colin and L. Jeanjean, Solutions for a quasilinear Schrödinger equations: A dual approach,, Nonlinear Anal., 56 (2004), 213. doi: 10.1016/j.na.2003.09.008. Google Scholar

[11]

M. Clapp and Y. H. Ding, Positive solutions of a Schrödinger equations with critical nonlinearity,, Z. Angew. Math. Phys., 55 (2004), 592. doi: 10.1007/s00033-004-1084-9. Google Scholar

[12]

M. del Pino and P. L. Felmer, Local mountain passes for semilinear elliptic problems in unbounded domains,, Calc. Var. PDE, 4 (1996), 121. doi: 10.1007/BF01189950. Google Scholar

[13]

Y. H. Ding and K. Tanaka, Multiplicity of positive solutions of a nonlinear Schrödinger equation,, Manuscript Math., 112 (2003), 109. doi: 10.1007/s00229-003-0397-x. Google Scholar

[14]

D. G. de Figueiredo and Y. H. Ding, Solutions of a nonlinear Schrödinger equation,, Discrete Contin. Dyn. System, 8 (2002), 563. doi: 10.3934/dcds.2002.8.563. Google Scholar

[15]

R. W. Hasse, A general method for the solution of nonlinear soliton and kink Schrödinger equation,, Z. Phys. B, 37 (1980), 83. doi: 10.1007/BF01325508. Google Scholar

[16]

S. Kurihara, Large-Amplitude quasi-solitons in superfluid films,, J. Phys. Soc. Japan, 50 (1981), 3262. doi: 10.1143/JPSJ.50.3262. Google Scholar

[17]

J. Q. Liu and Z. Q. Wang, Soliton solutions for quasilinear Schrödinger equations. I,, Proc. Amer. Math. Soc., 131 (2003), 441. doi: 10.1090/S0002-9939-02-06783-7. Google Scholar

[18]

J. Q. Liu, Y. Q. Wang and Z. Q. Wang, Soliton solutions for quasilinear Schrodinger equations. II,, J. Differential Equations, 187 (2003), 473. doi: 10.1016/S0022-0396(02)00064-5. Google Scholar

[19]

J. Q. Liu, Y. Q. 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

[20]

A. Nakamura, Damping and modification of exciton solitary waves,, J. Phys. Soc. Japan, 42 (1977), 1824. doi: 10.1143/JPSJ.42.1824. Google Scholar

[21]

J. M. B. do Ó, O. H. Miyagaki and S. H. M. 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

[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]

M. Porkolab and M. V. Goddman, Upper hybrid solitons and oscillating two-stream instabilities,, Phys. Fluids \textbf{19} (1976), 19 (1976), 872. doi: 10.1063/1.861553. Google Scholar

[24]

U. B. Severo, Existence results for quasilinear elliptic equations involving the p-Laplacian in the whole $\mathbbR^n$,, Electron. J. Differential Equations, 2008 (2008), 1. Google Scholar

[25]

U. B. Severo, Multiplicity of solutions for a class of quasilinear elliptic equations with concave and convex term in $\mathbbR$,, Electron. J. Qual. Theory Differ. Equ., 5 (2008), 1. Google Scholar

[26]

E. Séré, Existence of infinitely many homoclinic orbits in Hamiltonian systems,, Math. Z., 209 (1992), 27. doi: 10.1007/BF02570817. Google Scholar

[27]

M. Y. Yu and P. K. Shukla, On the formation of upper-hybrid solitons,, Plasma Phys., 19 (1977), 889. doi: 10.1088/0032-1028/19/9/008. Google Scholar

show all references

References:
[1]

C. O. Alves, Existence of multi-bump solutions for a class of quasilinear problems,, Adv. Nonlinear Stud., 6 (2006), 491. Google Scholar

[2]

C. O. Alves, D. C. de Morais Filho and M. A. S. Souto, Multiplicity of positive solutions for a class of problems with critical growth in $\mathbbR^N$,, Proc. Edinburgh Math. Soc., 52 (2009), 1. doi: 10.1017/S0013091507000028. Google Scholar

[3]

C. O. Alves and M. A. S. Souto, Multiplicity of positive solutions for a class of problems with exponential critical growth in $R^2$,, J. Differential Equations, 244 (2008), 1502. doi: 10.1016/j.jde.2007.09.007. Google Scholar

[4]

M. J. Alves, P. C. Carrião and O. H. Miyagaki, Soliton solutions to a class of quasilinear elliptic equations on $\mathbbR$,, Adv. Nonlinear Stud., 7 (2007), 579. Google Scholar

[5]

A. Ambrosetti and Z. Q. Wang, Positive solutions to a class of quasilinear elliptic equation on $\mathbbR$,, Discrete Contin. Dyn. Syst., 9 (2003), 55. doi: 10.3934/dcds.2003.9.55. Google Scholar

[6]

T. Bartsch and Z. Q. Wang, Existence and multiplicity results for some superlinear elliptic problems on $\mathbbR^N$,, Comm. Part. Diff. Eqs., 20 (1995), 1725. doi: 10.1080/03605309508821149. Google Scholar

[7]

T. Bartsch and Z. Q. Wang, Multiple positive solutions for a nonlinear Schrodinger equation,, Z. Angew Math. Phys., 51 (2000), 366. doi: 10.1007/PL00001511. Google Scholar

[8]

T. Bartsch, A. Pankov and Z. Q. Wang, Nonlinear Schrödinger equations with steep potential well,, Comm. Contemp. Math., 3 (2001), 549. doi: 10.1142/S0219199701000494. Google Scholar

[9]

J. Chen and B. Guo, Multiple nodal bound states for a quasilinear Schrödinger equations,, J. Math. Phys., 46 (2005). doi: 10.1063/1.2138045. Google Scholar

[10]

M. Colin and L. Jeanjean, Solutions for a quasilinear Schrödinger equations: A dual approach,, Nonlinear Anal., 56 (2004), 213. doi: 10.1016/j.na.2003.09.008. Google Scholar

[11]

M. Clapp and Y. H. Ding, Positive solutions of a Schrödinger equations with critical nonlinearity,, Z. Angew. Math. Phys., 55 (2004), 592. doi: 10.1007/s00033-004-1084-9. Google Scholar

[12]

M. del Pino and P. L. Felmer, Local mountain passes for semilinear elliptic problems in unbounded domains,, Calc. Var. PDE, 4 (1996), 121. doi: 10.1007/BF01189950. Google Scholar

[13]

Y. H. Ding and K. Tanaka, Multiplicity of positive solutions of a nonlinear Schrödinger equation,, Manuscript Math., 112 (2003), 109. doi: 10.1007/s00229-003-0397-x. Google Scholar

[14]

D. G. de Figueiredo and Y. H. Ding, Solutions of a nonlinear Schrödinger equation,, Discrete Contin. Dyn. System, 8 (2002), 563. doi: 10.3934/dcds.2002.8.563. Google Scholar

[15]

R. W. Hasse, A general method for the solution of nonlinear soliton and kink Schrödinger equation,, Z. Phys. B, 37 (1980), 83. doi: 10.1007/BF01325508. Google Scholar

[16]

S. Kurihara, Large-Amplitude quasi-solitons in superfluid films,, J. Phys. Soc. Japan, 50 (1981), 3262. doi: 10.1143/JPSJ.50.3262. Google Scholar

[17]

J. Q. Liu and Z. Q. Wang, Soliton solutions for quasilinear Schrödinger equations. I,, Proc. Amer. Math. Soc., 131 (2003), 441. doi: 10.1090/S0002-9939-02-06783-7. Google Scholar

[18]

J. Q. Liu, Y. Q. Wang and Z. Q. Wang, Soliton solutions for quasilinear Schrodinger equations. II,, J. Differential Equations, 187 (2003), 473. doi: 10.1016/S0022-0396(02)00064-5. Google Scholar

[19]

J. Q. Liu, Y. Q. 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

[20]

A. Nakamura, Damping and modification of exciton solitary waves,, J. Phys. Soc. Japan, 42 (1977), 1824. doi: 10.1143/JPSJ.42.1824. Google Scholar

[21]

J. M. B. do Ó, O. H. Miyagaki and S. H. M. 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

[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]

M. Porkolab and M. V. Goddman, Upper hybrid solitons and oscillating two-stream instabilities,, Phys. Fluids \textbf{19} (1976), 19 (1976), 872. doi: 10.1063/1.861553. Google Scholar

[24]

U. B. Severo, Existence results for quasilinear elliptic equations involving the p-Laplacian in the whole $\mathbbR^n$,, Electron. J. Differential Equations, 2008 (2008), 1. Google Scholar

[25]

U. B. Severo, Multiplicity of solutions for a class of quasilinear elliptic equations with concave and convex term in $\mathbbR$,, Electron. J. Qual. Theory Differ. Equ., 5 (2008), 1. Google Scholar

[26]

E. Séré, Existence of infinitely many homoclinic orbits in Hamiltonian systems,, Math. Z., 209 (1992), 27. doi: 10.1007/BF02570817. Google Scholar

[27]

M. Y. Yu and P. K. Shukla, On the formation of upper-hybrid solitons,, Plasma Phys., 19 (1977), 889. doi: 10.1088/0032-1028/19/9/008. Google Scholar

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