Ground state solutions for quasilinear stationary Schrödinger equations with critical growth

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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

Abstract
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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.
[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. 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.
[4]	H.Berestycki and P. L. Lions, Nonlinear scalar field equations, I - existence of a ground state,, Arch. Rat. Mech. Analysis, 82 (1983), 313. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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.
[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. 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. 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. doi: 10.1007/s00526-009-0299-1.
[20]	N. Trudinger, On imbedding into Orlicz spaces and some applications,, J. Math. Mech., 17 (1967), 473.

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.
[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. 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.
[4]	H.Berestycki and P. L. Lions, Nonlinear scalar field equations, I - existence of a ground state,, Arch. Rat. Mech. Analysis, 82 (1983), 313. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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.
[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. 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. 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. doi: 10.1007/s00526-009-0299-1.
[20]	N. Trudinger, On imbedding into Orlicz spaces and some applications,, J. Math. Mech., 17 (1967), 473.

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