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Positive solutions for quasilinear Schrödinger equations with critical growth and potential vanishing at infinity
1. | Department of Mathematics, Huazhong Normal University, Wuhan 430079 |
2. | Department of Mathematics, Huazhong Normal University, Wuhan, 430079, China |
References:
[1] |
Claudianor O. Alves and Marco A. S. Souto, Existence of solutions for a class of nonlinear Schrödinger equations with potentials vanishing at infinitly, J. Differential Equations, 254 (2013), 1977-1991.
doi: 10.1016/j.jde.2012.11.013. |
[2] |
A. Ambrosetti, V. Felli and A. Malchiodi, Ground states of nonlinear Schrödinger equations with potentials vanishing at infinitly, J. Eur. Math. Soc., 7 (2005), 117-144.
doi: 10.4171/JEMS/24. |
[3] |
A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Func. Anal., 14 (1973), 349-381. |
[4] |
A. Ambrosetti and Z-Q. Wang, Nonlinear Schrödinger equations with vanishing and decaying potentials, Differential Integral Equations, 18 (2005), 1321-1332. |
[5] |
S. Bae, H. O. Choi and D. H. Pahk, Existence of nodal solutions of nonlinear elliptic equations, Proc. Roy. Soc. Edinburgh Sect., A 137 (2007), 1135-1155.
doi: 10.1017/S0308210505000727. |
[6] |
Denis Bonheure and Jean Van Schaftingen, Groundstates for the nonlinear Schrödinger equation with potentials vanishing at infinitly, Ann. Mat. Pura Appl., 189 (2010), 273-301.
doi: 10.1007/s10231-009-0109-6. |
[7] |
T. Bartsch and M. Willem, Infinitely many radial solutions of a semilinear elliptic problem on $R^N$, Arch. Ration. Mech. Anal., 124 (1993), 261-276.
doi: 10.1007/BF00953069. |
[8] |
F. G. Bass and N. N. Nasanov, Nonlinear electromagnetic-spin waves, Phys. Rep., 189 (1990), 165-223.
doi: 10.1016/0370-1573(90)90093-H. |
[9] |
H. Berestycki and P. L. Lions, Nonlinear scalar field equations, I: Existence of a ground state, Arch. Ration. Mech. Anal., 82 (1983), 313-345.
doi: 10.1007/BF00250555. |
[10] |
João M. Bezerra do Ó, Olímpio H. Miyagaki and Sérgio H. M. Soares, Soliton solutions for quasilinear Schrödinger equations with critical growth, J. Differential Equations, 248 (2010), 722-744.
doi: 10.1016/j.jde.2009.11.030. |
[11] |
H. Brandi, C. Manus, G. Mainfray, T. Lehner and G. Bonnaud, Relativistic and ponderomotive self-focusing of a laser beam in a radially inhomogeneous plasma, Phys. Fluids B, 5 (1993), 3539-3550.
doi: 10.1063/1.870756. |
[12] |
H. Brezis and E. Lieb, A relation between pointwise convergence of function and convergence of functional, Proc. Amer. Math. Soc., 88 (1983), 486-490.
doi: 10.2307/2044999. |
[13] |
H. Brézis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure. Appl. Math., 36 (1983), 437-477
doi: 10.1002/cpa.3160360405. |
[14] |
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. |
[15] |
L. Brüll and H. Lange, Solitary waves for quasilinear Schrödinger equations, Exposition. Math., 4 (1986), 279-288. |
[16] |
G. Cerami, S. Solimini and M. Struwe, Some existence results for superlinear elliptic boundary value problems involving critical exponents, J. Func. Anal., 69 (1986), 289-306.
doi: 10.1016/0022-1236(86)90094-7. |
[17] |
X. L. Chen and R. N. Sudan, Necessary and sufficient conditions for self-focusing of short ultraintense laser pulse in underdense plasma, Phys. Rev. Lett., 70 (1993), 2082-2085. |
[18] |
M. Colin and L. Jeanjean, Solutions for a quasilinear Schrödinger equation: A dual approach, Nonlinear Anal. TMA., 56 (2004), 213-226.
doi: 10.1016/j.na.2003.09.008. |
[19] |
S. Cuccagna, On instability of excited states of the nonloinear Schrödinger equation, Physica D, 238 (2009), 38-54.
doi: 10.1016/j.physd.2008.08.010. |
[20] |
A. De Bouard, N. Hayashi and J. Saut, Global existence of small solutions to a relativistic nonlinear Schrödinger equation, Comm. Math. Phys., 189 (1997), 73-105.
doi: 10.1007/s002200050191. |
[21] |
Y. Deng, S. Peng and J. Wang, Infinitely many sign-changing solutions for quasilinear Schrödinger equations in $R^N$, Commun. Math. Sci., 9 (2011), 859-878.
doi: 10.4310/CMS.2011.v9.n3.a9. |
[22] |
Y. Deng, S. Peng and J. Wang, Nodal soliton solutions for quasilinear Schrödinger equations with critical exponent, J. Math. Phys., 54 (2013) 1-28
doi: 10.1063/1.4774153. |
[23] |
A. Floer and A. Weinstein, 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. |
[24] |
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. |
[25] |
A. M. Kosevich, B. A. Ivanov and A. S. Kovalev, Magnetic solitons, \it Phys. Rep., 194 (1990), 117-238.
doi: 10.1016/0370-1573(90)90130-T. |
[26] |
S. Kurihara, Large-amplitude quasi-solitons in superfluid films, J. Phys. Soc. Japan, 50 (1981), 3262-3267. |
[27] |
E. Laedke, K. 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. |
[28] |
H. Lange, M. Poppenberg and H. Teismann, Nash-Moser methods for the solution of quasilinear Schrödinger equations, Comm. Partial Differential Equations, 24 (1999), 1399-1418.
doi: 10.1080/03605309908821469. |
[29] |
J. Liu, Y. Wang and Z. Wang, Soliton solutions for quasilinear Schrödinger equations. II, J. Differential Equations, 187 (2003), 473-493.
doi: 10.1016/S0022-0396(02)00064-5. |
[30] |
J. Liu, Y. Wang and Z. Wang, Solutions for quasilinear Schrödinger equations via the Nehari method, Comm. Partial Differential Equations, 29 (2004), 879-901.
doi: 10.1081/PDE-120037335. |
[31] |
J. Liu and Z. Wang, Soliton solutions for quasilinear Schrödinger equations. I., Proc. Amer. Math. Soc., 131 (2003), 441-448.
doi: 10.1090/S0002-9939-02-06783-7. |
[32] |
J. Liu and Z. Wang, Symmetric solutions to a modified nonlinear Schrödinger equation, Nonlinearity, 21 (2008), 121-133.
doi: 10.1088/0951-7715/21/1/007. |
[33] |
Xiang-Qing Liu, Jia-Quan Liu and Zhi Qiang Wang, Quasilinear elliptic equations with critical growth via pertubation method, Journal Differential Equations, 254 (2013), 102-124
doi: 10.1016/j.jde.2012.09.006. |
[34] |
S. Lorca and P. Ubilla, Symmetric and nonsymmetric solutions for an elliptic equation on $R^N$, Nonlinear Anal., 58 (2004), 961-968.
doi: 10.1016/j.na.2004.03.034. |
[35] |
V. G. Makhankov and V. K. Fedyanin, Nonlinear effects in quasi-one-dimensional models and condensed matter theory, Phys. Rep., 104 (1984), 1-86.
doi: 10.1016/0370-1573(84)90106-6. |
[36] |
A. Moameni, Existence of soliton solutions for a quasilinear Schrödinger equation involving critical exponent in $\mathbb{R}^{N}$, J. Differential Equations, 229 (2006), 570-587.
doi: 10.1016/j.jde.2006.07.001. |
[37] |
M. Musso, F. Pacard and J. Wei, Finite-energy sign-changing solutions with dihedral symmetry for the stationary nonlinear Schrödinger equation, J. Eur. Math. Soc., 14 (2012), 1923-1953.
doi: 10.4171/JEMS/351. |
[38] |
B. Opic and A. Kufner, Hardy-Type Inequalities, Pitman Res. Notes Math. Ser. vol. 219. Longman Scientific and Technical. Harlow, 1990. |
[39] |
M. Poppenberg, K. Schmitt and Z. Wang, On the existence of soliton solutions to quasilinear Schrödinger equations, Calc. Var. Partial Differential Equations, 14 (2002), 329-344.
doi: 10.1007/s005260100105. |
[40] |
P. Pucci and J. Serrin, A general variational identity, Indiana Univ. Math. J., 35 (1986), 681-703.
doi: 10.1512/iumj.1986.35.35036. |
[41] |
G. R. W. Quispel and H. W. Capel, Equation of motion for the Heisenberg spin chain, Phys. A, 110 (1982), 41-80.
doi: 10.1016/0378-4371(82)90104-2. |
[42] |
P. H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270-291.
doi: 10.1007/BF00946631. |
[43] |
B. Ritchie, Relativistic self-focusing and channel formation in laser-plasma interactions, Phys. Rev. E, 50 (1994), 687-689. |
[44] |
W. A. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55 (1977), 149-162. |
show all references
References:
[1] |
Claudianor O. Alves and Marco A. S. Souto, Existence of solutions for a class of nonlinear Schrödinger equations with potentials vanishing at infinitly, J. Differential Equations, 254 (2013), 1977-1991.
doi: 10.1016/j.jde.2012.11.013. |
[2] |
A. Ambrosetti, V. Felli and A. Malchiodi, Ground states of nonlinear Schrödinger equations with potentials vanishing at infinitly, J. Eur. Math. Soc., 7 (2005), 117-144.
doi: 10.4171/JEMS/24. |
[3] |
A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Func. Anal., 14 (1973), 349-381. |
[4] |
A. Ambrosetti and Z-Q. Wang, Nonlinear Schrödinger equations with vanishing and decaying potentials, Differential Integral Equations, 18 (2005), 1321-1332. |
[5] |
S. Bae, H. O. Choi and D. H. Pahk, Existence of nodal solutions of nonlinear elliptic equations, Proc. Roy. Soc. Edinburgh Sect., A 137 (2007), 1135-1155.
doi: 10.1017/S0308210505000727. |
[6] |
Denis Bonheure and Jean Van Schaftingen, Groundstates for the nonlinear Schrödinger equation with potentials vanishing at infinitly, Ann. Mat. Pura Appl., 189 (2010), 273-301.
doi: 10.1007/s10231-009-0109-6. |
[7] |
T. Bartsch and M. Willem, Infinitely many radial solutions of a semilinear elliptic problem on $R^N$, Arch. Ration. Mech. Anal., 124 (1993), 261-276.
doi: 10.1007/BF00953069. |
[8] |
F. G. Bass and N. N. Nasanov, Nonlinear electromagnetic-spin waves, Phys. Rep., 189 (1990), 165-223.
doi: 10.1016/0370-1573(90)90093-H. |
[9] |
H. Berestycki and P. L. Lions, Nonlinear scalar field equations, I: Existence of a ground state, Arch. Ration. Mech. Anal., 82 (1983), 313-345.
doi: 10.1007/BF00250555. |
[10] |
João M. Bezerra do Ó, Olímpio H. Miyagaki and Sérgio H. M. Soares, Soliton solutions for quasilinear Schrödinger equations with critical growth, J. Differential Equations, 248 (2010), 722-744.
doi: 10.1016/j.jde.2009.11.030. |
[11] |
H. Brandi, C. Manus, G. Mainfray, T. Lehner and G. Bonnaud, Relativistic and ponderomotive self-focusing of a laser beam in a radially inhomogeneous plasma, Phys. Fluids B, 5 (1993), 3539-3550.
doi: 10.1063/1.870756. |
[12] |
H. Brezis and E. Lieb, A relation between pointwise convergence of function and convergence of functional, Proc. Amer. Math. Soc., 88 (1983), 486-490.
doi: 10.2307/2044999. |
[13] |
H. Brézis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure. Appl. Math., 36 (1983), 437-477
doi: 10.1002/cpa.3160360405. |
[14] |
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. |
[15] |
L. Brüll and H. Lange, Solitary waves for quasilinear Schrödinger equations, Exposition. Math., 4 (1986), 279-288. |
[16] |
G. Cerami, S. Solimini and M. Struwe, Some existence results for superlinear elliptic boundary value problems involving critical exponents, J. Func. Anal., 69 (1986), 289-306.
doi: 10.1016/0022-1236(86)90094-7. |
[17] |
X. L. Chen and R. N. Sudan, Necessary and sufficient conditions for self-focusing of short ultraintense laser pulse in underdense plasma, Phys. Rev. Lett., 70 (1993), 2082-2085. |
[18] |
M. Colin and L. Jeanjean, Solutions for a quasilinear Schrödinger equation: A dual approach, Nonlinear Anal. TMA., 56 (2004), 213-226.
doi: 10.1016/j.na.2003.09.008. |
[19] |
S. Cuccagna, On instability of excited states of the nonloinear Schrödinger equation, Physica D, 238 (2009), 38-54.
doi: 10.1016/j.physd.2008.08.010. |
[20] |
A. De Bouard, N. Hayashi and J. Saut, Global existence of small solutions to a relativistic nonlinear Schrödinger equation, Comm. Math. Phys., 189 (1997), 73-105.
doi: 10.1007/s002200050191. |
[21] |
Y. Deng, S. Peng and J. Wang, Infinitely many sign-changing solutions for quasilinear Schrödinger equations in $R^N$, Commun. Math. Sci., 9 (2011), 859-878.
doi: 10.4310/CMS.2011.v9.n3.a9. |
[22] |
Y. Deng, S. Peng and J. Wang, Nodal soliton solutions for quasilinear Schrödinger equations with critical exponent, J. Math. Phys., 54 (2013) 1-28
doi: 10.1063/1.4774153. |
[23] |
A. Floer and A. Weinstein, 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. |
[24] |
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. |
[25] |
A. M. Kosevich, B. A. Ivanov and A. S. Kovalev, Magnetic solitons, \it Phys. Rep., 194 (1990), 117-238.
doi: 10.1016/0370-1573(90)90130-T. |
[26] |
S. Kurihara, Large-amplitude quasi-solitons in superfluid films, J. Phys. Soc. Japan, 50 (1981), 3262-3267. |
[27] |
E. Laedke, K. 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. |
[28] |
H. Lange, M. Poppenberg and H. Teismann, Nash-Moser methods for the solution of quasilinear Schrödinger equations, Comm. Partial Differential Equations, 24 (1999), 1399-1418.
doi: 10.1080/03605309908821469. |
[29] |
J. Liu, Y. Wang and Z. Wang, Soliton solutions for quasilinear Schrödinger equations. II, J. Differential Equations, 187 (2003), 473-493.
doi: 10.1016/S0022-0396(02)00064-5. |
[30] |
J. Liu, Y. Wang and Z. Wang, Solutions for quasilinear Schrödinger equations via the Nehari method, Comm. Partial Differential Equations, 29 (2004), 879-901.
doi: 10.1081/PDE-120037335. |
[31] |
J. Liu and Z. Wang, Soliton solutions for quasilinear Schrödinger equations. I., Proc. Amer. Math. Soc., 131 (2003), 441-448.
doi: 10.1090/S0002-9939-02-06783-7. |
[32] |
J. Liu and Z. Wang, Symmetric solutions to a modified nonlinear Schrödinger equation, Nonlinearity, 21 (2008), 121-133.
doi: 10.1088/0951-7715/21/1/007. |
[33] |
Xiang-Qing Liu, Jia-Quan Liu and Zhi Qiang Wang, Quasilinear elliptic equations with critical growth via pertubation method, Journal Differential Equations, 254 (2013), 102-124
doi: 10.1016/j.jde.2012.09.006. |
[34] |
S. Lorca and P. Ubilla, Symmetric and nonsymmetric solutions for an elliptic equation on $R^N$, Nonlinear Anal., 58 (2004), 961-968.
doi: 10.1016/j.na.2004.03.034. |
[35] |
V. G. Makhankov and V. K. Fedyanin, Nonlinear effects in quasi-one-dimensional models and condensed matter theory, Phys. Rep., 104 (1984), 1-86.
doi: 10.1016/0370-1573(84)90106-6. |
[36] |
A. Moameni, Existence of soliton solutions for a quasilinear Schrödinger equation involving critical exponent in $\mathbb{R}^{N}$, J. Differential Equations, 229 (2006), 570-587.
doi: 10.1016/j.jde.2006.07.001. |
[37] |
M. Musso, F. Pacard and J. Wei, Finite-energy sign-changing solutions with dihedral symmetry for the stationary nonlinear Schrödinger equation, J. Eur. Math. Soc., 14 (2012), 1923-1953.
doi: 10.4171/JEMS/351. |
[38] |
B. Opic and A. Kufner, Hardy-Type Inequalities, Pitman Res. Notes Math. Ser. vol. 219. Longman Scientific and Technical. Harlow, 1990. |
[39] |
M. Poppenberg, K. Schmitt and Z. Wang, On the existence of soliton solutions to quasilinear Schrödinger equations, Calc. Var. Partial Differential Equations, 14 (2002), 329-344.
doi: 10.1007/s005260100105. |
[40] |
P. Pucci and J. Serrin, A general variational identity, Indiana Univ. Math. J., 35 (1986), 681-703.
doi: 10.1512/iumj.1986.35.35036. |
[41] |
G. R. W. Quispel and H. W. Capel, Equation of motion for the Heisenberg spin chain, Phys. A, 110 (1982), 41-80.
doi: 10.1016/0378-4371(82)90104-2. |
[42] |
P. H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270-291.
doi: 10.1007/BF00946631. |
[43] |
B. Ritchie, Relativistic self-focusing and channel formation in laser-plasma interactions, Phys. Rev. E, 50 (1994), 687-689. |
[44] |
W. A. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55 (1977), 149-162. |
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