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Well-posedness and scattering for fourth order nonlinear Schrödinger type equations at the scaling critical regularity
A class of generalized quasilinear Schrödinger equations
1. | Department of Mathematics, South China University of Technology, Guangzhou 510640 |
2. | School of Mathematics, South China University of Technology, Guangzhou 510640, China |
References:
[1] |
S. Adachia and T. Watanable, G-invariant positive solutions for a quasilinear Schrödinger equation, Adv. Diff. Eqns., 16 (2011), 289-324. |
[2] |
S. Adachia and T. Watanabeb, Uniqueness of the ground state solutions of quasilinear Schrödinger equations, Nonl. Anal. TMA., 75 (2012), 819-833. |
[3] |
A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381. |
[4] |
H. Berestycki and P. L. Lions, Nonlinear scalar field equations I, Arch. Rational Mech. Anal., 82 (1983), 313-346.
doi: 10.1007/BF00250555. |
[5] |
H. Brézis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponent, Comm. Pure Appl. Math., 36 (1983), 437-477.
doi: 10.1002/cpa.3160360405. |
[6] |
L. Brizhik, A. Eremko, B. Piette and W. J. Zakrzewski, Static solutions of a D-dimensional modified nonlinear Schrödinger equation, Nonlinearity, 16 (2003), 1481-1497.
doi: 10.1088/0951-7715/16/4/317. |
[7] |
M. Colin and L. Jeanjean, Solutions for a quasilinear Schrödinger equations: A dual approach, Nonl. Anal. TMA., 56 (2004), 213-226.
doi: 10.1016/j.na.2003.09.008. |
[8] |
A. De Bouard, N. Hayashi and J. C. Saut, Global existence of small solutions to a relativistic nonlinear Schrödinger equation, Comm. Math. Phys., 189 (1997), 73-105.
doi: 10.1007/s002200050191. |
[9] |
J. M. do Ó, O. Miyagaki and H. Olimpio, Soliton solutions for quasilinear Schrödinger equations with critical growth, J. Differential Equations, 248 (2010), 722-744.
doi: 10.1016/j.jde.2009.11.030. |
[10] |
B. Hartmann and W. Zakzewski, Electrons on hexagonal lattices and applications to nanotubes, Phys. Rev., 68 (2003), 1-9. |
[11] |
R. W. Hasse, A general method for the solution of nonlinear soliton and kink Schrödinger equations, Z. Phys., 37 (1980), 83-87.
doi: 10.1007/BF01325508. |
[12] |
L. Jeanjean and K. Tanaka, A remark on least energy solutions in $\mathbbR^N$, Proc. Amer. Math. Soc., 131 (2003), 2399-2408. |
[13] |
A. M. Kosevich, B. A. Ivanov and A. S. Kovalev, Magnetic solitons, Phys. Rep., 194 (1990), 117-238. |
[14] |
S. Kurihura, Large-amplitude quasi-solitons in superfluid films, J. Phys. Soc. Japan., 50 (1981), 3262-3267. |
[15] |
E. W. Laedke, K. H. Spatschek and L. Stenflo, Evolution theorem for a class of perturbed envelope soliton solutions, J. Math. Phys., 24 (1983), 2764-2769.
doi: 10.1063/1.525675. |
[16] |
A. G. Litvak and A. M. Sergeev, One dimensional collapse of plasma waves, JETP Lett., 27 (1978), 517-520. |
[17] |
J. Q. Liu and Z. Q. Wang, Soliton solutions for quasilinear Schrödinger equations I, Proc. Amer. Math. Soc., 131 (2002), 441-448. |
[18] |
J. Q. Liu, Y. Q, 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. |
[19] |
J. Q. Liu, Y. Q. Wang and Z. Q. Wang, Solutions for quasilinear Schrödinger equations via the Nehari Method, Commun. Partial Differ. Equ., 29 (2004), 879-901.
doi: 10.1081/PDE-120037335. |
[20] |
V. G. Makhankov and V. K. Fedyanin, Non-linear effects in quasi-one-dimensional models of condensed matter theory, Phys. Rep., 104 (1984), 1-86.
doi: 10.1016/0370-1573(84)90106-6. |
[21] |
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. |
[22] |
G. R. W. Quispel and H. W. Capel, Equation of motion for the Heisenberg spin chain, Physica A, 110 (1982), 41-80.
doi: 10.1016/0378-4371(82)90104-2. |
[23] |
D. Ruiz and G. Siciliano, Existence of ground states for a modified nonlinear Schrödinger equation, Nonlinearity, 23 (2010), 1221-1233.
doi: 10.1088/0951-7715/23/5/011. |
[24] |
Y. T. Shen and X. K. Guo, The positive solutions of degenerate variational problem and degenerate elliptic equations, Chin. J. Cont. Mathematics, 14 (1993), 157-166. |
[25] |
Y. T. Shen and Y. J. Wang, Soliton solutions for generalized quasilinear Schrödinger equations, Nonl. Anal. TMA., 80 (2013), 194-201.
doi: 10.1016/j.na.2012.10.005. |
[26] |
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. |
[27] |
Y. J. Wang, Y. M. Zhang and Y. T. Shen, Multiple solutions for quasilinear Schrödinger equations involving critical exponent, Appl. Math. Comp., 216 (2010), 849-856.
doi: 10.1016/j.amc.2010.01.091. |
[28] |
Y. J. Wang and W. M, Zou, Bound states to critical quasilinear Schrödinger equations, Nonl. Diff. Equa. Appl., 19 (2012), 19-47. |
[29] |
J. Yang, Y. J. Wang and A. A. Abdelgadir, Soliton solutions for quasilinear Schrödinger equations, J. Math. Phys, 54 (2013), 071502. |
show all references
References:
[1] |
S. Adachia and T. Watanable, G-invariant positive solutions for a quasilinear Schrödinger equation, Adv. Diff. Eqns., 16 (2011), 289-324. |
[2] |
S. Adachia and T. Watanabeb, Uniqueness of the ground state solutions of quasilinear Schrödinger equations, Nonl. Anal. TMA., 75 (2012), 819-833. |
[3] |
A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381. |
[4] |
H. Berestycki and P. L. Lions, Nonlinear scalar field equations I, Arch. Rational Mech. Anal., 82 (1983), 313-346.
doi: 10.1007/BF00250555. |
[5] |
H. Brézis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponent, Comm. Pure Appl. Math., 36 (1983), 437-477.
doi: 10.1002/cpa.3160360405. |
[6] |
L. Brizhik, A. Eremko, B. Piette and W. J. Zakrzewski, Static solutions of a D-dimensional modified nonlinear Schrödinger equation, Nonlinearity, 16 (2003), 1481-1497.
doi: 10.1088/0951-7715/16/4/317. |
[7] |
M. Colin and L. Jeanjean, Solutions for a quasilinear Schrödinger equations: A dual approach, Nonl. Anal. TMA., 56 (2004), 213-226.
doi: 10.1016/j.na.2003.09.008. |
[8] |
A. De Bouard, N. Hayashi and J. C. Saut, Global existence of small solutions to a relativistic nonlinear Schrödinger equation, Comm. Math. Phys., 189 (1997), 73-105.
doi: 10.1007/s002200050191. |
[9] |
J. M. do Ó, O. Miyagaki and H. Olimpio, Soliton solutions for quasilinear Schrödinger equations with critical growth, J. Differential Equations, 248 (2010), 722-744.
doi: 10.1016/j.jde.2009.11.030. |
[10] |
B. Hartmann and W. Zakzewski, Electrons on hexagonal lattices and applications to nanotubes, Phys. Rev., 68 (2003), 1-9. |
[11] |
R. W. Hasse, A general method for the solution of nonlinear soliton and kink Schrödinger equations, Z. Phys., 37 (1980), 83-87.
doi: 10.1007/BF01325508. |
[12] |
L. Jeanjean and K. Tanaka, A remark on least energy solutions in $\mathbbR^N$, Proc. Amer. Math. Soc., 131 (2003), 2399-2408. |
[13] |
A. M. Kosevich, B. A. Ivanov and A. S. Kovalev, Magnetic solitons, Phys. Rep., 194 (1990), 117-238. |
[14] |
S. Kurihura, Large-amplitude quasi-solitons in superfluid films, J. Phys. Soc. Japan., 50 (1981), 3262-3267. |
[15] |
E. W. Laedke, K. H. Spatschek and L. Stenflo, Evolution theorem for a class of perturbed envelope soliton solutions, J. Math. Phys., 24 (1983), 2764-2769.
doi: 10.1063/1.525675. |
[16] |
A. G. Litvak and A. M. Sergeev, One dimensional collapse of plasma waves, JETP Lett., 27 (1978), 517-520. |
[17] |
J. Q. Liu and Z. Q. Wang, Soliton solutions for quasilinear Schrödinger equations I, Proc. Amer. Math. Soc., 131 (2002), 441-448. |
[18] |
J. Q. Liu, Y. Q, 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. |
[19] |
J. Q. Liu, Y. Q. Wang and Z. Q. Wang, Solutions for quasilinear Schrödinger equations via the Nehari Method, Commun. Partial Differ. Equ., 29 (2004), 879-901.
doi: 10.1081/PDE-120037335. |
[20] |
V. G. Makhankov and V. K. Fedyanin, Non-linear effects in quasi-one-dimensional models of condensed matter theory, Phys. Rep., 104 (1984), 1-86.
doi: 10.1016/0370-1573(84)90106-6. |
[21] |
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. |
[22] |
G. R. W. Quispel and H. W. Capel, Equation of motion for the Heisenberg spin chain, Physica A, 110 (1982), 41-80.
doi: 10.1016/0378-4371(82)90104-2. |
[23] |
D. Ruiz and G. Siciliano, Existence of ground states for a modified nonlinear Schrödinger equation, Nonlinearity, 23 (2010), 1221-1233.
doi: 10.1088/0951-7715/23/5/011. |
[24] |
Y. T. Shen and X. K. Guo, The positive solutions of degenerate variational problem and degenerate elliptic equations, Chin. J. Cont. Mathematics, 14 (1993), 157-166. |
[25] |
Y. T. Shen and Y. J. Wang, Soliton solutions for generalized quasilinear Schrödinger equations, Nonl. Anal. TMA., 80 (2013), 194-201.
doi: 10.1016/j.na.2012.10.005. |
[26] |
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. |
[27] |
Y. J. Wang, Y. M. Zhang and Y. T. Shen, Multiple solutions for quasilinear Schrödinger equations involving critical exponent, Appl. Math. Comp., 216 (2010), 849-856.
doi: 10.1016/j.amc.2010.01.091. |
[28] |
Y. J. Wang and W. M, Zou, Bound states to critical quasilinear Schrödinger equations, Nonl. Diff. Equa. Appl., 19 (2012), 19-47. |
[29] |
J. Yang, Y. J. Wang and A. A. Abdelgadir, Soliton solutions for quasilinear Schrödinger equations, J. Math. Phys, 54 (2013), 071502. |
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