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January  2017, 37(1): 77-103. doi: 10.3934/dcds.2017004

Nodal solutions for a generalized quasilinear Schrödinger equation with critical exponents

Department of Mathematics, Huazhong Normal University, Wuhan 430079, China

Received  March 2016 Revised  April 2016 Published  November 2016

Fund Project: This work was supported by the Natural Science Foundation of China (11371160,11328101) and the Program for Changjiang Scholars and Innovative Research Team in University (#IRT13066).

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 $k ≥ 0$.

Citation: Kun Cheng, Yinbin Deng. Nodal solutions for a generalized quasilinear Schrödinger equation with critical exponents. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 77-103. doi: 10.3934/dcds.2017004
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.  Google Scholar

[2]

S. BaeH. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

J. M. Bezerra do ÓO. 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.  Google Scholar

[6]

G. BianchiJ. 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.  Google Scholar

[7]

A. De BouardN. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

L. Brüll and H. Lange, Solitary waves for quasilinear Schrödinger equations, Exposition. Math., 4 (1986), 279-288.  doi: 10.1515/ans-2009-0303.  Google Scholar

[11]

D. Cao and X. Zhu, on the existence and nodal character of semilinear elliptic equations, Acta. Math. Sci., 8 (1988), 345-359.   Google Scholar

[12]

G. CeramiS. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[18]

Y. DengS. 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.  Google Scholar

[19]

Y. DengS. 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.  Google Scholar

[20]

W. H. Fleming, A selection-migration model in population genetic, J. Math. Bio., 2 (1975), 219-233.  doi: 10.1007/BF00277151.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

P. L. Kelley, Self focusing of optical beams, Phys. Rev. Lett., 15 (1965), 1005-1008.  doi: 10.1109/IQEC.2005.1561150.  Google Scholar

[24]

A. M. KosevichB. A. Ivanov and A. S. Kovalev, Magnetic solitons, Phys. Rep., 194 (1990), 117-238.  doi: 10.1016/0370-1573(90)90130-T.  Google Scholar

[25]

S. Kurihara, Large-amplitude quasi-solitons in superfluid films, J. Phys. Soc. Japan, 50 (1981), 3262-3267.  doi: 10.1143/JPSJ.50.3262.  Google Scholar

[26]

E. LaedkeK. 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.  Google Scholar

[27]

H. LangeM. 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.  Google Scholar

[28]

J. LiuY. 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.  Google Scholar

[29]

J. LiuY. 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.  Google Scholar

[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.  Google Scholar

[31]

X. LiuJ. 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.  Google Scholar

[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.  Google Scholar

[33]

C. Miranda, Un'osservazione su un teorema di Brouwer, Boll. Un. Mat. Ital., 3 (1940), 5-7.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[36]

M. PoppenbergK. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[41]

W. A. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55 (1977), 149-162.  doi: 10.1007/bf01626517.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

S. BaeH. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

J. M. Bezerra do ÓO. 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.  Google Scholar

[6]

G. BianchiJ. 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.  Google Scholar

[7]

A. De BouardN. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

L. Brüll and H. Lange, Solitary waves for quasilinear Schrödinger equations, Exposition. Math., 4 (1986), 279-288.  doi: 10.1515/ans-2009-0303.  Google Scholar

[11]

D. Cao and X. Zhu, on the existence and nodal character of semilinear elliptic equations, Acta. Math. Sci., 8 (1988), 345-359.   Google Scholar

[12]

G. CeramiS. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[18]

Y. DengS. 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.  Google Scholar

[19]

Y. DengS. 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.  Google Scholar

[20]

W. H. Fleming, A selection-migration model in population genetic, J. Math. Bio., 2 (1975), 219-233.  doi: 10.1007/BF00277151.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

P. L. Kelley, Self focusing of optical beams, Phys. Rev. Lett., 15 (1965), 1005-1008.  doi: 10.1109/IQEC.2005.1561150.  Google Scholar

[24]

A. M. KosevichB. A. Ivanov and A. S. Kovalev, Magnetic solitons, Phys. Rep., 194 (1990), 117-238.  doi: 10.1016/0370-1573(90)90130-T.  Google Scholar

[25]

S. Kurihara, Large-amplitude quasi-solitons in superfluid films, J. Phys. Soc. Japan, 50 (1981), 3262-3267.  doi: 10.1143/JPSJ.50.3262.  Google Scholar

[26]

E. LaedkeK. 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.  Google Scholar

[27]

H. LangeM. 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.  Google Scholar

[28]

J. LiuY. 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.  Google Scholar

[29]

J. LiuY. 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.  Google Scholar

[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.  Google Scholar

[31]

X. LiuJ. 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.  Google Scholar

[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.  Google Scholar

[33]

C. Miranda, Un'osservazione su un teorema di Brouwer, Boll. Un. Mat. Ital., 3 (1940), 5-7.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[36]

M. PoppenbergK. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[41]

W. A. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55 (1977), 149-162.  doi: 10.1007/bf01626517.  Google Scholar

[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.  Google Scholar

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