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Ground state solutions for quasilinear stationary Schrödinger equations with critical growth
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 |
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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