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On the structure of solutions of nonlinear hyperbolic systems of conservation laws
Solutions for quasilinear Schrödinger equations with critical Sobolev-Hardy exponents
1. | Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, PO Box 71010, Wuhan 430071E01103, China |
2. | Department of Mathematical Sciences, Tsinghua University, Beijing 100084e/pjp, China |
3. | Department of Mathematics, South China University of Technology, Guangzhou 510640, China |
References:
[1] |
C. O. Alves, J. M. do Ó and O. Miyagaki, On nonlinear perturbations of a periodic elliptic problem in $\mathbb{R}^2$ involving critical growth, Nonlinear Anal., 56 (2004), 781-791.
doi: 10.1016/j.na.2003.06.003. |
[2] |
M. Colin and L. Jeanjean, Solutions for a quasilinear Schrödinger equations: A dual approach, Nonlinear Anal., 56 (2004), 213-226.
doi: 10.1016/j.na.2003.09.008. |
[3] |
J. M. do Ó, O. Miyagaki and S. 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. |
[4] |
J. M. do Ó and U. Severo, Quasilinear Schrödinger equations involving concave and convex nonlinearities, Comm. Pure and Applied Anal., 8 (2009), 621-644.
doi: 10.3934/cpaa.2009.8.621. |
[5] |
A. Floer and A. Weisntein, Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential, J. Funct. Anal., 69 (1986), 397-408.
doi: 10.1016/0022-1236(86)90096-0. |
[6] |
N. Ghoussoub and C. Yuan, Multiple solutions for quasi-linear PDEs involving the critical Sobolev and Hardy exponents, Trans. Amer. Math. Soc., 352 (2000), 5703-5743.
doi: 10.1090/S0002-9947-00-02560-5. |
[7] |
R. W. Hasse, A general method for the solution of nonlinear soliton and kink Schrödinger equations, Z. Phys. B, 37 (1980), 83-87.
doi: 10.1007/BF01325508. |
[8] |
D. Sh. Kang, Solutions for semilinear elliptic problems with critical Sobolev-Hardy exponents in $\mathbb{R}^N2$, Nonlinear Anal., 66 (2007), 241–252.
doi: 10.1016/j.na.2005.11.028. |
[9] |
A. M. Kosevich, B. A. Ivanov and A. S. Kovalev, Magnetic solitons, Phys. Rep., 194 (1990), 117-238.
doi: 10.1016/0370-1573(90)90130-T. |
[10] |
S. Kurihura, Large-amplitude quasi-solitons in superfluid films, J. Phys. Soc. Japan, 50 (1981), 3262-3267.
doi: 10.1143/jpsj.50.3262. |
[11] |
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. |
[12] |
A. G. Litvak and A. M. Sergeev, One dimensional collapse of plasma waves, JETP Lett., 27 (1978), 517-520.
doi: 0021-3640778/2710-0517. |
[13] |
J. Q. Liu, Y. Q. 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. |
[14] |
J. Q. Liu and Z. Q. Wang, Soliton solutions for quasilinear Schrödinger equations I, Proc. Amer. Math. Soc., 131 (2002), 441-448.
doi: 10.1090/S0002-9939-02-06783-7. |
[15] |
J. Q. Liu and Z. Q. Wang, Soliton solutions for quasilinear Schrödinger equations II, J. Differential Equations, 187 (2003), 473-493.
doi: 10.1016/s0022-0396(02)0064-5. |
[16] |
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. |
[17] |
J. Mawhin and M. Willem, "Critical Point Theory and Hamiltonian Systems," Springer-Verlag, 1989. |
[18] |
O. H. Miyagaki, On a class of semilinear elliptic problems in $\mathbb{R}^N2$ with critical growth, Nonlinear Anal., 29 (1997), 773-781.
doi: 10.1016/s0362-546x(96)00087-9. |
[19] |
A. Moameni, Existence of soliton solutions for a quasilinear Schrödinger equation involving critical exponent in $\mathbb{R}^N2$, J. Differential Equations, 229 (2006), 570-587.
doi: 10.1016/j.jde.2006.07.001. |
[20] |
A. Moameni, On the existence of standing wave solutions to quasilinear Schrödinger equations, Nonlinearity, 19 (2006), 937-957.
doi: 10.1088/0951-7715/19/4/009. |
[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] |
S. Takeno and S. Homma, Classical planar Heisenberg ferromagnet, complex scalar fields and nonlinear excitations, Progr. Theoret. Phys., 65 (1981), 172-189.
doi: 10.1143/PTP.65.172. |
[24] |
P. H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z angew Math. Phy., 43 (1992), 272-291.
doi: 10.1007/BF00946631. |
[25] |
Y. J. Wang, J. Yang and Y. M. Zhang, Quasilinear elliptic equations involving the N-Laplacian with critical exponential growth in $\mathbb{R}^N2$, Nonlinear Anal., 71 (2009), 6157-6169.
doi: 10.1016/j.na.2009.06.006. |
[26] |
Y. J. Wang, Y. M. Zhang and Y. T. Shen, Multiple solutions for quasilinear Schröinger equations involving critical exponent, Applied Mathematics and Computation, 216 (2010), 849-856.
doi: 10.1016/j.amc.2010.01.091. |
show all references
References:
[1] |
C. O. Alves, J. M. do Ó and O. Miyagaki, On nonlinear perturbations of a periodic elliptic problem in $\mathbb{R}^2$ involving critical growth, Nonlinear Anal., 56 (2004), 781-791.
doi: 10.1016/j.na.2003.06.003. |
[2] |
M. Colin and L. Jeanjean, Solutions for a quasilinear Schrödinger equations: A dual approach, Nonlinear Anal., 56 (2004), 213-226.
doi: 10.1016/j.na.2003.09.008. |
[3] |
J. M. do Ó, O. Miyagaki and S. 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. |
[4] |
J. M. do Ó and U. Severo, Quasilinear Schrödinger equations involving concave and convex nonlinearities, Comm. Pure and Applied Anal., 8 (2009), 621-644.
doi: 10.3934/cpaa.2009.8.621. |
[5] |
A. Floer and A. Weisntein, Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential, J. Funct. Anal., 69 (1986), 397-408.
doi: 10.1016/0022-1236(86)90096-0. |
[6] |
N. Ghoussoub and C. Yuan, Multiple solutions for quasi-linear PDEs involving the critical Sobolev and Hardy exponents, Trans. Amer. Math. Soc., 352 (2000), 5703-5743.
doi: 10.1090/S0002-9947-00-02560-5. |
[7] |
R. W. Hasse, A general method for the solution of nonlinear soliton and kink Schrödinger equations, Z. Phys. B, 37 (1980), 83-87.
doi: 10.1007/BF01325508. |
[8] |
D. Sh. Kang, Solutions for semilinear elliptic problems with critical Sobolev-Hardy exponents in $\mathbb{R}^N2$, Nonlinear Anal., 66 (2007), 241–252.
doi: 10.1016/j.na.2005.11.028. |
[9] |
A. M. Kosevich, B. A. Ivanov and A. S. Kovalev, Magnetic solitons, Phys. Rep., 194 (1990), 117-238.
doi: 10.1016/0370-1573(90)90130-T. |
[10] |
S. Kurihura, Large-amplitude quasi-solitons in superfluid films, J. Phys. Soc. Japan, 50 (1981), 3262-3267.
doi: 10.1143/jpsj.50.3262. |
[11] |
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. |
[12] |
A. G. Litvak and A. M. Sergeev, One dimensional collapse of plasma waves, JETP Lett., 27 (1978), 517-520.
doi: 0021-3640778/2710-0517. |
[13] |
J. Q. Liu, Y. Q. 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. |
[14] |
J. Q. Liu and Z. Q. Wang, Soliton solutions for quasilinear Schrödinger equations I, Proc. Amer. Math. Soc., 131 (2002), 441-448.
doi: 10.1090/S0002-9939-02-06783-7. |
[15] |
J. Q. Liu and Z. Q. Wang, Soliton solutions for quasilinear Schrödinger equations II, J. Differential Equations, 187 (2003), 473-493.
doi: 10.1016/s0022-0396(02)0064-5. |
[16] |
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. |
[17] |
J. Mawhin and M. Willem, "Critical Point Theory and Hamiltonian Systems," Springer-Verlag, 1989. |
[18] |
O. H. Miyagaki, On a class of semilinear elliptic problems in $\mathbb{R}^N2$ with critical growth, Nonlinear Anal., 29 (1997), 773-781.
doi: 10.1016/s0362-546x(96)00087-9. |
[19] |
A. Moameni, Existence of soliton solutions for a quasilinear Schrödinger equation involving critical exponent in $\mathbb{R}^N2$, J. Differential Equations, 229 (2006), 570-587.
doi: 10.1016/j.jde.2006.07.001. |
[20] |
A. Moameni, On the existence of standing wave solutions to quasilinear Schrödinger equations, Nonlinearity, 19 (2006), 937-957.
doi: 10.1088/0951-7715/19/4/009. |
[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] |
S. Takeno and S. Homma, Classical planar Heisenberg ferromagnet, complex scalar fields and nonlinear excitations, Progr. Theoret. Phys., 65 (1981), 172-189.
doi: 10.1143/PTP.65.172. |
[24] |
P. H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z angew Math. Phy., 43 (1992), 272-291.
doi: 10.1007/BF00946631. |
[25] |
Y. J. Wang, J. Yang and Y. M. Zhang, Quasilinear elliptic equations involving the N-Laplacian with critical exponential growth in $\mathbb{R}^N2$, Nonlinear Anal., 71 (2009), 6157-6169.
doi: 10.1016/j.na.2009.06.006. |
[26] |
Y. J. Wang, Y. M. Zhang and Y. T. Shen, Multiple solutions for quasilinear Schröinger equations involving critical exponent, Applied Mathematics and Computation, 216 (2010), 849-856.
doi: 10.1016/j.amc.2010.01.091. |
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