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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 & Applied Analysis, 2015, 14 (6) : 2487-2508. doi: 10.3934/cpaa.2015.14.2487
References:
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S. Bae, H. O. Choi and D. H. Pahk, Existence of nodal solutions of nonlinear elliptic equations,, \emph{Proc. Roy. Soc. Edinburgh Sect., A 137 (2007), 1135. doi: 10.1017/S0308210505000727. 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,, \emph{Phys. Fluids B, 5 (1993), 3539. Google Scholar

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D. Cao and X. Zhu, On the existence and nodal character of semilinear elliptic equations,, \emph{Acta. Math. Sci., 8 (1988), 345. Google Scholar

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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,, \emph{Phys. Rev. Lett., 70 (1993), 2082. Google Scholar

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G. Cerami, S. Solimini and M. Struwe, Some existence results for superlinear elliptic boundary value problems involving critical exponents,, \emph{J. Func. Anal., 69 (1986), 289. doi: 10.1016/0022-1236(86)90094-7. Google Scholar

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Y. Deng, S. Peng and J. Wang, Infinitely many sign-changing solutions for quasilinear Schrödinger equations in $\R^N$,, \emph{Commun. Math. Sci., 9 (2011), 859. doi: 10.4310/CMS.2011.v9.n3.a9. Google Scholar

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S. Kurihara, Large-amplitude quasi-solitons in superfluid films,, \emph{J. Phys. Soc. Japan, 50 (1981), 3262. Google Scholar

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Xiang-Qing Liu, Jia-Quan Liu and Zhi Qiang Wang, Quasilinear elliptic equations with critical growth via pertubation method,, \emph{Journal Differential Equations, 254 (2013), 102. doi: 10.1016/j.jde.2012.09.006. Google Scholar

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J. Liu and Z. Wang, Soliton solutions for quasilinear Schrödinger equations. I.,, \emph{Proc. Amer. Math. Soc., 131 (2003), 441. doi: 10.1090/S0002-9939-02-06783-7. Google Scholar

[25]

J. Liu and Z. Wang, Symmetric solutions to a modified nonlinear Schrödinger equation,, \emph{Nonlinearity, 21 (2008), 121. doi: 10.1088/0951-7715/21/1/007. Google Scholar

[26]

J. Liu, Y. Wang and Z. Wang, Soliton solutions for quasilinear Schrödinger equations. II,, \emph{J. Differential Equations, 187 (2003), 473. doi: 10.1016/S0022-0396(02)00064-5. Google Scholar

[27]

J. Liu, Y. Wang and Z. Wang, Solutions for quasilinear Schrödinger equations via the Nehari method,, \emph{Comm. Partial Differential Equations, 29 (2004), 879. doi: 10.1081/PDE-120037335. Google Scholar

[28]

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

[29]

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

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A. Moameni, Existence of soliton solutions for a quasilinear Schrödinger equation involving critical exponent in $\R^N$,, \emph{J. Differential Equations, 229 (2006), 570. doi: 10.1016/j.jde.2006.07.001. Google Scholar

[31]

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

[32]

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

[33]

M. Poppenberg, K. Schmitt and Z. Wang, On the existence of soliton solutions to quasilinear Schrödinger equations,, \emph{Calc. Var. Partial Differential Equations, 14 (2002), 329. doi: 10.1007/s005260100105. Google Scholar

[34]

B. Ritchie, Relativistic self-focusing and channel formation in laser-plasma interactions,, \emph{Phys. Rev. E, 50 (1994), 687. Google Scholar

[35]

Y. Shen and Y. Wang, Soliton solutions for generalized quasilinear Schrodinger equations,, \emph{Nonlinear Anal. TMA., 80 (2013), 194. doi: 10.1016/j.na.2012.10.005. Google Scholar

[36]

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

[37]

W. A. Strauss, Existence of solitary waves in higher dimensions,, \emph{Comm. Math. Phys., 55 (1977), 149. Google Scholar

[38]

T. Weth, Energy bounds for entire nodal solutions of autonomous superlinear equations,, \emph{Calc. Var. Partial Differential Equations, 27 (2006), 421. 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,, \emph{J. Func. Anal., 14 (1973), 349. Google Scholar

[2]

S. Bae, H. O. Choi and D. H. Pahk, Existence of nodal solutions of nonlinear elliptic equations,, \emph{Proc. Roy. Soc. Edinburgh Sect., A 137 (2007), 1135. doi: 10.1017/S0308210505000727. Google Scholar

[3]

T. Bartsch and M. Willem, Infinitely many radial solutions of a semilinear elliptic problem on $\R^N$,, \emph{Arch. Ration. Mech. Anal., 124 (1993), 261. doi: 10.1007/BF00953069. Google Scholar

[4]

T. Bartsch and Z. Tang, Multibump solutions of nonlinear Schrodinger equations with steep potential well and indefinite potential,, \emph{Discrete Contin. Dyn. Syst.}, 33 (2012), 7. Google Scholar

[5]

G. Bianchi, J. Chabrowski and A. Szulkin, On symmetric solutions of an elliptic equation with a nonlinearity involving critical Sobolev exponent,, \emph{Nonlinear Anal. TMA, 25 (1995), 41. doi: 10.1016/0362-546X(94)E0070-W. Google Scholar

[6]

João M. Bezerra do Ó, Olímpio H. Miyagaki and Sérgio H. M. Soares, Soliton solutions for quasilinear Schrödinger equations with critical growth,, \emph{J. Differential Equations, 248 (2010), 722. doi: 10.1016/j.jde.2009.11.030. Google Scholar

[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,, \emph{Phys. Fluids B, 5 (1993), 3539. Google Scholar

[8]

H. Brezis and E. Lieb, A relation between pointwise convergence of function and convergence of functional,, \emph{Proc. Amer. Math. Soc., 88 (1983), 486. doi: 10.2307/2044999. Google Scholar

[9]

D. Cao and X. Zhu, On the existence and nodal character of semilinear elliptic equations,, \emph{Acta. Math. Sci., 8 (1988), 345. Google Scholar

[10]

X. L. Chen and R. N. Sudan, Necessary and sufficient conditions for self-focusing of short ultraintense laser pulse in underdense plasma,, \emph{Phys. Rev. Lett., 70 (1993), 2082. Google Scholar

[11]

G. Cerami, S. Solimini and M. Struwe, Some existence results for superlinear elliptic boundary value problems involving critical exponents,, \emph{J. Func. Anal., 69 (1986), 289. doi: 10.1016/0022-1236(86)90094-7. Google Scholar

[12]

M. Colin and L. Jeanjean, Solutions for a quasilinear Schrödinger equation: A dual approach,, \emph{Nonlinear Anal. TMA., 56 (2004), 213. doi: 10.1016/j.na.2003.09.008. Google Scholar

[13]

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

[14]

A. De Bouard, N. Hayashi and J. Saut, Global existence of small solutions to a relativistic nonlinear Schrödinger equation,, \emph{Comm. Math. Phys., 189 (1997), 73. doi: 10.1007/s002200050191. Google Scholar

[15]

Y. Deng, The existence and nodal character of solutions in $\R ^N$ for semilinear elliptic equations involving critical Sobolev exponents,, \emph{Acta. Math. Sci., 9 (1989), 385. Google Scholar

[16]

Y. Deng, S. Peng and J. Wang, Infinitely many sign-changing solutions for quasilinear Schrödinger equations in $\R^N$,, \emph{Commun. Math. Sci., 9 (2011), 859. doi: 10.4310/CMS.2011.v9.n3.a9. Google Scholar

[17]

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

[18]

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

[19]

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

[20]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order,, {Springer-Verlag, (1998). Google Scholar

[21]

S. Kurihara, Large-amplitude quasi-solitons in superfluid films,, \emph{J. Phys. Soc. Japan, 50 (1981), 3262. Google Scholar

[22]

Xiang-Qing Liu, Jia-Quan Liu and Zhi Qiang Wang, Quasilinear elliptic equations with critical growth via pertubation method,, \emph{Journal Differential Equations, 254 (2013), 102. doi: 10.1016/j.jde.2012.09.006. Google Scholar

[23]

E. Laedke, K. Spatschek and L. Stenflo, Evolution theorem for a class of perturbed envelope soliton solutions,, \emph{J. Math. Phys., 24 (1983), 2764. doi: 10.1063/1.525675. Google Scholar

[24]

J. Liu and Z. Wang, Soliton solutions for quasilinear Schrödinger equations. I.,, \emph{Proc. Amer. Math. Soc., 131 (2003), 441. doi: 10.1090/S0002-9939-02-06783-7. Google Scholar

[25]

J. Liu and Z. Wang, Symmetric solutions to a modified nonlinear Schrödinger equation,, \emph{Nonlinearity, 21 (2008), 121. doi: 10.1088/0951-7715/21/1/007. Google Scholar

[26]

J. Liu, Y. Wang and Z. Wang, Soliton solutions for quasilinear Schrödinger equations. II,, \emph{J. Differential Equations, 187 (2003), 473. doi: 10.1016/S0022-0396(02)00064-5. Google Scholar

[27]

J. Liu, Y. Wang and Z. Wang, Solutions for quasilinear Schrödinger equations via the Nehari method,, \emph{Comm. Partial Differential Equations, 29 (2004), 879. doi: 10.1081/PDE-120037335. Google Scholar

[28]

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

[29]

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

[30]

A. Moameni, Existence of soliton solutions for a quasilinear Schrödinger equation involving critical exponent in $\R^N$,, \emph{J. Differential Equations, 229 (2006), 570. doi: 10.1016/j.jde.2006.07.001. Google Scholar

[31]

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

[32]

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

[33]

M. Poppenberg, K. Schmitt and Z. Wang, On the existence of soliton solutions to quasilinear Schrödinger equations,, \emph{Calc. Var. Partial Differential Equations, 14 (2002), 329. doi: 10.1007/s005260100105. Google Scholar

[34]

B. Ritchie, Relativistic self-focusing and channel formation in laser-plasma interactions,, \emph{Phys. Rev. E, 50 (1994), 687. Google Scholar

[35]

Y. Shen and Y. Wang, Soliton solutions for generalized quasilinear Schrodinger equations,, \emph{Nonlinear Anal. TMA., 80 (2013), 194. doi: 10.1016/j.na.2012.10.005. Google Scholar

[36]

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

[37]

W. A. Strauss, Existence of solitary waves in higher dimensions,, \emph{Comm. Math. Phys., 55 (1977), 149. Google Scholar

[38]

T. Weth, Energy bounds for entire nodal solutions of autonomous superlinear equations,, \emph{Calc. Var. Partial Differential Equations, 27 (2006), 421. doi: 10.1007/s00526-006-0015-3. Google Scholar

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