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January  2013, 12(1): 99-116. doi: 10.3934/cpaa.2013.12.99

Ground state solutions for quasilinear stationary Schrödinger equations with critical growth

Marco A. S. Souto 1, and  Sérgio H. M. Soares 2,

1. 

Universidade Federal de Campina Grande, Departamento de Matemática e Estatística, 58429-000 Campina Grande, PB, Brazil
2. 

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

Received  August 2010 Revised  July 2012 Published  September 2012

We establish the existence of ground state solution for quasilinear Schrödinger equations involving critical growth. The method used here is minimizing the gradient integral norm in a manifold defined by integrals involving the primitive of the nonlinearity function.
Keywords: ground state solution, Mountain pass, variational methods..
Mathematics Subject Classification: Primary: 35J20, 35J60; Secondary: 35J1.
Citation: Marco A. S. Souto, Sérgio H. M. Soares. Ground state solutions for quasilinear stationary Schrödinger equations with critical growth. Communications on Pure and Applied Analysis, 2013, 12 (1) : 99-116. doi: 10.3934/cpaa.2013.12.99
References:
[1]

S. Adachi and T. Watanabe, $G$-invariant positive solutions for a quasilinear Schrödinger equation, Adv. Differential Equations, 16 (2011), 289-324.

[2]

C. O. Alves, M. S. Montenegro and M. A. S. Souto, Existence of a ground state solution for a nonlinear scalar field equation with critical growth, Calc. Var. Partial Differential Equations, 43 (2012), 537-554. doi: 10.1007/s00526-011-0422-y.

[3]

H. Berestycki and O. Kavian, Equations de Champs scalaires euclidiens non linéaires dans le plan, C. R. Acad. Sci. Paris Ser. I Math, 297 (1984), 307-310.

[4]

H.Berestycki and P. L. Lions, Nonlinear scalar field equations, I - existence of a ground state, Arch. Rat. Mech. Analysis, 82 (1983), 313-346. doi: 10.1007/BF00250555.

[5]

L. Brizhik, A. Eremko, B. Piette and W. J. Zakrzewski, Electron self-trapping in a discrete two-dimensional lattice, Physica D: Nonlinear Phenomena, 159 (2001), 71-90. doi: 10.1016/S0167-2789(01)00332-3.

[6]

D. M. Cao, Nontrivial solution of semilinear elliptic equation with critical exponent $\mathbbR^2$, Comm. Part. Diff. Equations, 17 (1992), 407-435. doi: 10.1080/03605309208820848.

[7]

S. Coleman, V. Glazer and A. Martin, Action minima among solutions to a class of Euclidean scalar field equations, Comm. Math. Phys, 58 (1978), 211-221. doi: 10.1007/BF01609421.

[8]

M. Colin, L. Jeanjean and M. Squassina, Stability and instability results for standing waves of quasi-linear Schrödinger equations, Nonlinearity, 23 (2010), 1353-1385. doi: 10.1088/0951-7715/23/6/006.

[9]

M. Colin and L. Jeanjean, Solutions for a quasilinear Schrödinger equation: a dual approach, Nonlinear Anal., 56 (2004), 213-226. doi: 10.1016/j.na.2003.09.008.

[10]

J. M. do Ó, 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.

[11]

J. M. do Ó, O. H. Miyagaki and S. H. M. Soares, Soliton solutions for quasilinear Schrödinger equations: the critical exponential case, Nonlinear Anal., 67 (2007), 3357-3372. doi: 10.1016/j.na.2006.10.018.

[12]

L. Jeanjean, L. and K. Tanaka, A Remark on least energy solutions in $\mathbbR^N$, Proc. Amer. Mathematical Society, 131 (2003), 2399-2408. doi: 10.1090/S0002-9939-02-06821-1.

[13]

P. L. Lions, The concentration-compactness principle in the calculus of variations. The limit case. I, Rev. Mat. Iberoamericana, 1 (1985), 145-201. doi: 10.4171/RMI/6.

[14]

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.

[15]

J. Liu and Z.-Q. Wang, Soliton solutions for quasilinear Schrödinger equations. I, Proc. Amer. Math. Soc., 131 (2003), 441-448.

[16]

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.

[17]

J. Moser, A sharp form of an inequality by N. Trudinger,, Indiana Univ. Math. J., 20 (): 1077.  doi: 10.1512/iumj.1971.20.20101.

[18]

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.

[19]

E. A. 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.

[20]

N. Trudinger, On imbedding into Orlicz spaces and some applications, J. Math. Mech., 17 (1967), 473-484.

show all references

References:
[1]

S. Adachi and T. Watanabe, $G$-invariant positive solutions for a quasilinear Schrödinger equation, Adv. Differential Equations, 16 (2011), 289-324.

[2]

C. O. Alves, M. S. Montenegro and M. A. S. Souto, Existence of a ground state solution for a nonlinear scalar field equation with critical growth, Calc. Var. Partial Differential Equations, 43 (2012), 537-554. doi: 10.1007/s00526-011-0422-y.

[3]

H. Berestycki and O. Kavian, Equations de Champs scalaires euclidiens non linéaires dans le plan, C. R. Acad. Sci. Paris Ser. I Math, 297 (1984), 307-310.

[4]

H.Berestycki and P. L. Lions, Nonlinear scalar field equations, I - existence of a ground state, Arch. Rat. Mech. Analysis, 82 (1983), 313-346. doi: 10.1007/BF00250555.

[5]

L. Brizhik, A. Eremko, B. Piette and W. J. Zakrzewski, Electron self-trapping in a discrete two-dimensional lattice, Physica D: Nonlinear Phenomena, 159 (2001), 71-90. doi: 10.1016/S0167-2789(01)00332-3.

[6]

D. M. Cao, Nontrivial solution of semilinear elliptic equation with critical exponent $\mathbbR^2$, Comm. Part. Diff. Equations, 17 (1992), 407-435. doi: 10.1080/03605309208820848.

[7]

S. Coleman, V. Glazer and A. Martin, Action minima among solutions to a class of Euclidean scalar field equations, Comm. Math. Phys, 58 (1978), 211-221. doi: 10.1007/BF01609421.

[8]

M. Colin, L. Jeanjean and M. Squassina, Stability and instability results for standing waves of quasi-linear Schrödinger equations, Nonlinearity, 23 (2010), 1353-1385. doi: 10.1088/0951-7715/23/6/006.

[9]

M. Colin and L. Jeanjean, Solutions for a quasilinear Schrödinger equation: a dual approach, Nonlinear Anal., 56 (2004), 213-226. doi: 10.1016/j.na.2003.09.008.

[10]

J. M. do Ó, 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.

[11]

J. M. do Ó, O. H. Miyagaki and S. H. M. Soares, Soliton solutions for quasilinear Schrödinger equations: the critical exponential case, Nonlinear Anal., 67 (2007), 3357-3372. doi: 10.1016/j.na.2006.10.018.

[12]

L. Jeanjean, L. and K. Tanaka, A Remark on least energy solutions in $\mathbbR^N$, Proc. Amer. Mathematical Society, 131 (2003), 2399-2408. doi: 10.1090/S0002-9939-02-06821-1.

[13]

P. L. Lions, The concentration-compactness principle in the calculus of variations. The limit case. I, Rev. Mat. Iberoamericana, 1 (1985), 145-201. doi: 10.4171/RMI/6.

[14]

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.

[15]

J. Liu and Z.-Q. Wang, Soliton solutions for quasilinear Schrödinger equations. I, Proc. Amer. Math. Soc., 131 (2003), 441-448.

[16]

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.

[17]

J. Moser, A sharp form of an inequality by N. Trudinger,, Indiana Univ. Math. J., 20 (): 1077.  doi: 10.1512/iumj.1971.20.20101.

[18]

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.

[19]

E. A. 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.

[20]

N. Trudinger, On imbedding into Orlicz spaces and some applications, J. Math. Mech., 17 (1967), 473-484.

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