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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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

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

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

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

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