Nodal solutions for a generalized quasilinear Schrödinger equation with critical exponents

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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 and Continuous Dynamical Systems, 2017, 37 (1) : 77-103. doi: 10.3934/dcds.2017004

References:

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[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 ℝ^N, 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 ℝ^N 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 ℝ^N, 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 ℝ^N, 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 ℝ^N, 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 ℝ^N 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 ℝ^N, 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 ℝ^N, 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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