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On the Smoluchowski-Kramers approximation for SPDEs and its interplay with large deviations and long time behavior
Nodal solutions for a generalized quasilinear Schrödinger equation with critical exponents
Department of Mathematics, Huazhong Normal University, Wuhan 430079, China |
This paper is concerned with constructing nodal radial solutions for generalized quasilinear Schrödinger equations in $\mathbb{R}^N$ with critical growth which arise from plasma physics, fluid mechanics, as well as the self-channeling of a high-power ultashort laser in matter. We find the critical exponents for a generalized quasilinear Schrödinger equations and obtain the existence of sign-changing solution with k nodes for any given integer $k ≥ 0$.
References:
[1] |
A. Ambrosetti and P. H. Rabinowitz,
Dual variational methods in critical point theory and applications, J. Func. Anal., 14 (1973), 349-381.
doi: 10.1016/0022-1236(73)90051-7. |
[2] |
S. Bae, H. O. Choi and D. H. Pahk,
Existence of nodal solutions of nonlinear elliptic equations, Proc. Roy. Soc. Edinburgh Sect., 137 (2007), 1135-1155.
doi: 10.1017/S0308210505000727. |
[3] |
T. Bartsch and M. Willem,
Infinitely many radial solutions of a semilinear elliptic problem on ℝN, Arch. Ration. Mech. Anal., 124 (1993), 261-276.
doi: 10.1007/BF00953069. |
[4] |
F. G. Bass and N. N. Nasanov,
Nonlinear electromagnetic-spin waves, Phys. Rep., 169 (1990), 165-223.
doi: 10.1016/0370-1573(90)90093-H. |
[5] |
J. M. Bezerra do Ó, O. H. Miyagaki and S. 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. |
[6] |
G. Bianchi, J. Chabrowski and A. Szulkin,
On symmetric solutions of an elliptic equation with a nonlinearity involving critical Sobolev exponent, Nonlinear Anal. TMA., 25 (1995), 41-59.
doi: 10.1016/0362-546X(94)E0070-W. |
[7] |
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. |
[8] |
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 1 (1994), p968.
doi: 10.1063/1.870756. |
[9] |
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: 10.1090/S0002-9939-1983-0699419-3. |
[10] |
L. Brüll and H. Lange,
Solitary waves for quasilinear Schrödinger equations, Exposition. Math., 4 (1986), 279-288.
doi: 10.1515/ans-2009-0303. |
[11] |
D. Cao and X. Zhu,
on the existence and nodal character of semilinear elliptic equations, Acta. Math. Sci., 8 (1988), 345-359.
|
[12] |
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. |
[13] |
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.
doi: 10.1103/PhysRevLett.70.2082. |
[14] |
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. |
[15] |
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. |
[16] |
Y. Deng,
The existence and nodal character of solutions in ℝN for semilinear elliptic equations involving critical Sobolev exponents, Acta. Math. Sci., 9 (1989), 385-402.
|
[17] |
Y. Deng, S. Peng and J. Wang, Nodal soliton solutions for generalized quasilinear Schrödinger equations J. Math. Phys. 55 (2014), 051501, 16pp.
doi: 10.1063/1.4874108. |
[18] |
Y. Deng, S. Peng and J. Wang,
Infinitely many sign-changing solutions for quasilinear Schrödinger equations in ℝN, Commun. Math. Sci., 9 (2011), 859-878.
doi: 10.4310/CMS.2011.v9.n3.a9. |
[19] |
Y. Deng, S. Peng and S. Yan,
Critical exponents and solitary wave solutions for generalized quaslinear Schrödinger equations, J. Differential Equations, 260 (2016), 1228-1262.
doi: 10.1016/j.jde.2015.09.021. |
[20] |
W. H. Fleming,
A selection-migration model in population genetic, J. Math. Bio., 2 (1975), 219-233.
doi: 10.1007/BF00277151. |
[21] |
D. Gilbarg and N. S. Trudinger,
Elliptic Partial Differential Equations Of Second Order Springer-Verlag, Berlin, 2001.
doi: 10.1007/978-3-642-61798-0. |
[22] |
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. |
[23] |
P. L. Kelley,
Self focusing of optical beams, Phys. Rev. Lett., 15 (1965), 1005-1008.
doi: 10.1109/IQEC.2005.1561150. |
[24] |
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. |
[25] |
S. Kurihara,
Large-amplitude quasi-solitons in superfluid films, J. Phys. Soc. Japan, 50 (1981), 3262-3267.
doi: 10.1143/JPSJ.50.3262. |
[26] |
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. |
[27] |
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. |
[28] |
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. |
[29] |
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. |
[30] |
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. |
[31] |
X. Liu, J. Liu and Z. Wang,
Quasilinear elliptic equations with critical growth via pertubation method, J. Differential Equations, 254 (2013), 102-124.
doi: 10.1016/j.jde.2012.09.006. |
[32] |
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. |
[33] |
C. Miranda,
Un'osservazione su un teorema di Brouwer, Boll. Un. Mat. Ital., 3 (1940), 5-7.
|
[34] |
A. Moameni,
Existence of soliton solutions for a quasilinear Schrödinger equation involving critical exponent in ℝN, J. Differential Equations, 229 (2006), 570-587.
doi: 10.1016/j.jde.2006.07.001. |
[35] |
Z. Nehari,
Characteristic values associated with a class of non-linear second-order differential equations, Acta Math., 105 (1961), 141-175.
doi: 10.1007/BF02559588. |
[36] |
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. |
[37] |
P. Pucci and J. Serrin,
A general variational identity, Indiana Univ. Math. J., 35 (1986), 681-703.
doi: 10.1512/iumj.1986.35.35036. |
[38] |
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. |
[39] |
B. Ritchie,
Relativistic self-focusing and channel formation in laser-plasma interactions, Phys. Rev. E, 50 (1994), 687-689.
doi: 10.1103/PhysRevE.50.R687. |
[40] |
Y. Shen and Y. Wang,
Soliton solutions for generalized quasilinear Schrodinger equations, Nonlinear Anal. TMA., 80 (2013), 194-201.
doi: 10.1016/j.na.2012.10.005. |
[41] |
W. A. Strauss,
Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55 (1977), 149-162.
doi: 10.1007/bf01626517. |
[42] |
T. Weth,
Energy bounds for entire nodal solutions of autonomous superlinear equations, Calc. Var. Partial Differential Equations, 27 (2006), 421-437.
doi: 10.1007/s00526-006-0015-3. |
show all references
References:
[1] |
A. Ambrosetti and P. H. Rabinowitz,
Dual variational methods in critical point theory and applications, J. Func. Anal., 14 (1973), 349-381.
doi: 10.1016/0022-1236(73)90051-7. |
[2] |
S. Bae, H. O. Choi and D. H. Pahk,
Existence of nodal solutions of nonlinear elliptic equations, Proc. Roy. Soc. Edinburgh Sect., 137 (2007), 1135-1155.
doi: 10.1017/S0308210505000727. |
[3] |
T. Bartsch and M. Willem,
Infinitely many radial solutions of a semilinear elliptic problem on ℝN, Arch. Ration. Mech. Anal., 124 (1993), 261-276.
doi: 10.1007/BF00953069. |
[4] |
F. G. Bass and N. N. Nasanov,
Nonlinear electromagnetic-spin waves, Phys. Rep., 169 (1990), 165-223.
doi: 10.1016/0370-1573(90)90093-H. |
[5] |
J. M. Bezerra do Ó, O. H. Miyagaki and S. 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. |
[6] |
G. Bianchi, J. Chabrowski and A. Szulkin,
On symmetric solutions of an elliptic equation with a nonlinearity involving critical Sobolev exponent, Nonlinear Anal. TMA., 25 (1995), 41-59.
doi: 10.1016/0362-546X(94)E0070-W. |
[7] |
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. |
[8] |
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 1 (1994), p968.
doi: 10.1063/1.870756. |
[9] |
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: 10.1090/S0002-9939-1983-0699419-3. |
[10] |
L. Brüll and H. Lange,
Solitary waves for quasilinear Schrödinger equations, Exposition. Math., 4 (1986), 279-288.
doi: 10.1515/ans-2009-0303. |
[11] |
D. Cao and X. Zhu,
on the existence and nodal character of semilinear elliptic equations, Acta. Math. Sci., 8 (1988), 345-359.
|
[12] |
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. |
[13] |
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.
doi: 10.1103/PhysRevLett.70.2082. |
[14] |
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. |
[15] |
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. |
[16] |
Y. Deng,
The existence and nodal character of solutions in ℝN for semilinear elliptic equations involving critical Sobolev exponents, Acta. Math. Sci., 9 (1989), 385-402.
|
[17] |
Y. Deng, S. Peng and J. Wang, Nodal soliton solutions for generalized quasilinear Schrödinger equations J. Math. Phys. 55 (2014), 051501, 16pp.
doi: 10.1063/1.4874108. |
[18] |
Y. Deng, S. Peng and J. Wang,
Infinitely many sign-changing solutions for quasilinear Schrödinger equations in ℝN, Commun. Math. Sci., 9 (2011), 859-878.
doi: 10.4310/CMS.2011.v9.n3.a9. |
[19] |
Y. Deng, S. Peng and S. Yan,
Critical exponents and solitary wave solutions for generalized quaslinear Schrödinger equations, J. Differential Equations, 260 (2016), 1228-1262.
doi: 10.1016/j.jde.2015.09.021. |
[20] |
W. H. Fleming,
A selection-migration model in population genetic, J. Math. Bio., 2 (1975), 219-233.
doi: 10.1007/BF00277151. |
[21] |
D. Gilbarg and N. S. Trudinger,
Elliptic Partial Differential Equations Of Second Order Springer-Verlag, Berlin, 2001.
doi: 10.1007/978-3-642-61798-0. |
[22] |
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. |
[23] |
P. L. Kelley,
Self focusing of optical beams, Phys. Rev. Lett., 15 (1965), 1005-1008.
doi: 10.1109/IQEC.2005.1561150. |
[24] |
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. |
[25] |
S. Kurihara,
Large-amplitude quasi-solitons in superfluid films, J. Phys. Soc. Japan, 50 (1981), 3262-3267.
doi: 10.1143/JPSJ.50.3262. |
[26] |
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. |
[27] |
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. |
[28] |
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. |
[29] |
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. |
[30] |
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. |
[31] |
X. Liu, J. Liu and Z. Wang,
Quasilinear elliptic equations with critical growth via pertubation method, J. Differential Equations, 254 (2013), 102-124.
doi: 10.1016/j.jde.2012.09.006. |
[32] |
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. |
[33] |
C. Miranda,
Un'osservazione su un teorema di Brouwer, Boll. Un. Mat. Ital., 3 (1940), 5-7.
|
[34] |
A. Moameni,
Existence of soliton solutions for a quasilinear Schrödinger equation involving critical exponent in ℝN, J. Differential Equations, 229 (2006), 570-587.
doi: 10.1016/j.jde.2006.07.001. |
[35] |
Z. Nehari,
Characteristic values associated with a class of non-linear second-order differential equations, Acta Math., 105 (1961), 141-175.
doi: 10.1007/BF02559588. |
[36] |
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. |
[37] |
P. Pucci and J. Serrin,
A general variational identity, Indiana Univ. Math. J., 35 (1986), 681-703.
doi: 10.1512/iumj.1986.35.35036. |
[38] |
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. |
[39] |
B. Ritchie,
Relativistic self-focusing and channel formation in laser-plasma interactions, Phys. Rev. E, 50 (1994), 687-689.
doi: 10.1103/PhysRevE.50.R687. |
[40] |
Y. Shen and Y. Wang,
Soliton solutions for generalized quasilinear Schrodinger equations, Nonlinear Anal. TMA., 80 (2013), 194-201.
doi: 10.1016/j.na.2012.10.005. |
[41] |
W. A. Strauss,
Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55 (1977), 149-162.
doi: 10.1007/bf01626517. |
[42] |
T. Weth,
Energy bounds for entire nodal solutions of autonomous superlinear equations, Calc. Var. Partial Differential Equations, 27 (2006), 421-437.
doi: 10.1007/s00526-006-0015-3. |
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