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November  2015, 14(6): 2487-2508. doi: 10.3934/cpaa.2015.14.2487

Nodal solutions for a quasilinear Schrödinger equation with critical nonlinearity and non-square diffusion

Yinbin Deng 1, , Yi Li 2, and  Xiujuan Yan 3,

1. 

Department of Mathematics, Huazhong Normal University, Wuhan 430079
2. 

Department of Mathematics and Statistics, Wright State University, Dayton, OH 45435, USA
3. 

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

Received  June 2015 Revised  June 2015 Published  September 2015

This paper is concerned with a type of quasilinear Schrödinger equations of the form \begin{eqnarray} -\Delta u+V(x)u-p\Delta(|u|^{2p})|u|^{2p-2}u=\lambda|u|^{q-2}u+|u|^{2p2^{*}-2}u, \end{eqnarray} where $\lambda>0, N\ge3, 4p < q < 2p2^*, 2^*=\frac{2N}{N-2}, 1< p < +\infty$. For any given $k \ge 0$, by using a change of variables and Nehari minimization, we obtain a sign-changing minimizer with $k$ nodes.
Keywords: radial solutions, Quasilinear Schrödinger equations, nodal solutions..
Mathematics Subject Classification: 35J20, 35J62, 35Q5.
Citation: Yinbin Deng, Yi Li, Xiujuan Yan. Nodal solutions for a quasilinear Schrödinger equation with critical nonlinearity and non-square diffusion. Communications on Pure and Applied Analysis, 2015, 14 (6) : 2487-2508. doi: 10.3934/cpaa.2015.14.2487
References:
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A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Func. Anal., 14 (1973), 349-381.

[2]

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.

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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]

T. Bartsch and Z. Tang, Multibump solutions of nonlinear Schrodinger equations with steep potential well and indefinite potential, Discrete Contin. Dyn. Syst., 33, (2012), 7-26,

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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.

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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.

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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, 5 (1993), 3539-3550.

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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.

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

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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.

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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.

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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.

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M. Conti, L. Merizzi and S. Terracini, Radial solutions of superlinear equations on $\R^N$. I. A global variational approach, Arch. Ration. Mech. Anal., 153 (2000), 291-316. doi: 10.1007/s002050050015.

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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.

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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.

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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.

[17]

Y. Deng, S. Peng and J. Wang, Nodal soliton solutions for quasilinear Schrödinger equations with critical exponent, Journal of Mathematical Physics, 54 (2013), 011504. doi: 10.1063/1.4774153.

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Y. Deng and W. Shuai, Positive solutions for quasilinear Schrodinger equations with critical growth and potential vanishing at infinity, Commun. Pure Appl. Anal., 13 (2014), 2273-2287. doi: 10.3934/cpaa.2014.13.2273.

[19]

P. Felmer and C. Torres, Radial symmetry of ground states for a regional fractional nonlinear Schrodinger equation, Commun. Pure Appl. Anal., 13 (2014), 2395-2406. doi: 10.3934/cpaa.2014.13.2395.

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D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin-New York, 1998.

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S. Kurihara, Large-amplitude quasi-solitons in superfluid films, J. Phys. Soc. Japan, 50 (1981), 3262-3267.

[22]

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.

[23]

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.

[24]

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.

[25]

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.

[26]

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.

[27]

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.

[28]

X. Q. Liu and J. Q. Liu, Quasilinear elliptic equations via perturbation meathod, Proc. Amer. Math. Soc. 141 (2013), 253-263. doi: 10.1090/S0002-9939-2012-11293-6.

[29]

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

[30]

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.

[31]

A. Moameni, Soliton solutions for quasilinear Schrodinger equations involving supercritical exponent in $\R^N$, Commun. Pure Appl. Anal., 7 (2007), 89-105. doi: 10.3934/cpaa.2008.7.89.

[32]

Z. Nehari, Characteristic values associated with a class of non-linear second-order differential equations, Acta Math., 105 (1961), 141-175.

[33]

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.

[34]

B. Ritchie, Relativistic self-focusing and channel formation in laser-plasma interactions, Phys. Rev. E, 50 (1994), 687-689.

[35]

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.

[36]

Marco A. S. Souto and Sergio H. M. Soares, Ground state solutions for quasilinear stationary Schrodinger equations with critical growth, Commun. Pure Appl. Anal., 12 (2012), 99-116. doi: 10.3934/cpaa.2013.12.99.

[37]

W. A. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55 (1977), 149-162.

[38]

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.

[2]

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.

[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]

T. Bartsch and Z. Tang, Multibump solutions of nonlinear Schrodinger equations with steep potential well and indefinite potential, Discrete Contin. Dyn. Syst., 33, (2012), 7-26,

[5]

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.

[6]

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.

[7]

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.

[8]

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.

[9]

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

[10]

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.

[11]

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.

[12]

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.

[13]

M. Conti, L. Merizzi and S. Terracini, Radial solutions of superlinear equations on $\R^N$. I. A global variational approach, Arch. Ration. Mech. Anal., 153 (2000), 291-316. doi: 10.1007/s002050050015.

[14]

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.

[15]

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.

[16]

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.

[17]

Y. Deng, S. Peng and J. Wang, Nodal soliton solutions for quasilinear Schrödinger equations with critical exponent, Journal of Mathematical Physics, 54 (2013), 011504. doi: 10.1063/1.4774153.

[18]

Y. Deng and W. Shuai, Positive solutions for quasilinear Schrodinger equations with critical growth and potential vanishing at infinity, Commun. Pure Appl. Anal., 13 (2014), 2273-2287. doi: 10.3934/cpaa.2014.13.2273.

[19]

P. Felmer and C. Torres, Radial symmetry of ground states for a regional fractional nonlinear Schrodinger equation, Commun. Pure Appl. Anal., 13 (2014), 2395-2406. doi: 10.3934/cpaa.2014.13.2395.

[20]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin-New York, 1998.

[21]

S. Kurihara, Large-amplitude quasi-solitons in superfluid films, J. Phys. Soc. Japan, 50 (1981), 3262-3267.

[22]

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.

[23]

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.

[24]

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.

[25]

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.

[26]

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.

[27]

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.

[28]

X. Q. Liu and J. Q. Liu, Quasilinear elliptic equations via perturbation meathod, Proc. Amer. Math. Soc. 141 (2013), 253-263. doi: 10.1090/S0002-9939-2012-11293-6.

[29]

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

[30]

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.

[31]

A. Moameni, Soliton solutions for quasilinear Schrodinger equations involving supercritical exponent in $\R^N$, Commun. Pure Appl. Anal., 7 (2007), 89-105. doi: 10.3934/cpaa.2008.7.89.

[32]

Z. Nehari, Characteristic values associated with a class of non-linear second-order differential equations, Acta Math., 105 (1961), 141-175.

[33]

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.

[34]

B. Ritchie, Relativistic self-focusing and channel formation in laser-plasma interactions, Phys. Rev. E, 50 (1994), 687-689.

[35]

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.

[36]

Marco A. S. Souto and Sergio H. M. Soares, Ground state solutions for quasilinear stationary Schrodinger equations with critical growth, Commun. Pure Appl. Anal., 12 (2012), 99-116. doi: 10.3934/cpaa.2013.12.99.

[37]

W. A. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55 (1977), 149-162.

[38]

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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