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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

Kun Cheng and  Yinbin Deng

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 $k ≥ 0$.

Keywords: Quasilinear Schrödinger equations, critical exponents, nodal solutions, mountain pass lemma, constrained minimization problem.
Mathematics Subject Classification: Primary:35Q55;Secondary:35J62, 35J2.
Citation: Kun Cheng, Yinbin Deng. Nodal solutions for a generalized quasilinear Schrödinger equation with critical exponents. Discrete & Continuous Dynamical Systems, 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. 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.  Google Scholar

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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

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[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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

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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. 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.  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. 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.  Google Scholar

[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.  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. Kosevich, B. 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. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  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. 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.  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. 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.  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. 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.  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 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.  Google Scholar

[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.  Google Scholar

[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.  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. 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.  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. 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.  Google Scholar

[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.  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. Kosevich, B. 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. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  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. 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.  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. 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.  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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