May  2016, 15(3): 853-870. doi: 10.3934/cpaa.2016.15.853

A class of generalized quasilinear Schrödinger equations

1. 

Department of Mathematics, South China University of Technology, Guangzhou 510640

2. 

School of Mathematics, South China University of Technology, Guangzhou 510640, China

Received  June 2015 Revised  November 2015 Published  February 2015

We establish the existence of nontrivial solutions for the following quasilinear Schrödinger equation with critical Sobolev exponent: \begin{eqnarray} -\Delta u+V(x) u-\Delta [l(u^2)]l'(u^2)u= \lambda u^{\alpha2^*-1}+h(u),\ \ x\in \mathbb{R}^N, \end{eqnarray} where $V(x):\mathbb{R}^N\rightarrow \mathbb{R}$ is a given potential and $l,h$ are real functions, $\lambda\geq 0$, $\alpha>1$, $2^*=2N/(N-2)$, $N\geq 3$. Our results cover two physical models $l(s)=s^{\frac{\alpha}{2}}$ and $l(s) = (1+s)^{\frac{\alpha}{2}}$ with $\alpha\geq 3/2$.
Citation: Yaotian Shen, Youjun Wang. A class of generalized quasilinear Schrödinger equations. Communications on Pure & Applied Analysis, 2016, 15 (3) : 853-870. doi: 10.3934/cpaa.2016.15.853
References:
[1]

S. Adachia and T. Watanable, G-invariant positive solutions for a quasilinear Schrödinger equation,, \emph{Adv. Diff. Eqns.}, 16 (2011), 289. Google Scholar

[2]

S. Adachia and T. Watanabeb, Uniqueness of the ground state solutions of quasilinear Schrödinger equations,, \emph{Nonl. Anal. TMA.}, 75 (2012), 819. Google Scholar

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A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications,, \emph{J. Funct. Anal.}, 14 (1973), 349. Google Scholar

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H. Berestycki and P. L. Lions, Nonlinear scalar field equations I,, \emph{Arch. Rational Mech. Anal.}, 82 (1983), 313. doi: 10.1007/BF00250555. Google Scholar

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H. Brézis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponent,, \emph{Comm. Pure Appl. Math.}, 36 (1983), 437. doi: 10.1002/cpa.3160360405. Google Scholar

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L. Brizhik, A. Eremko, B. Piette and W. J. Zakrzewski, Static solutions of a D-dimensional modified nonlinear Schrödinger equation,, \emph{Nonlinearity}, 16 (2003), 1481. doi: 10.1088/0951-7715/16/4/317. Google Scholar

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

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A. De Bouard, N. Hayashi and J. C. 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

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J. M. do Ó, O. Miyagaki and H. Olimpio, 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

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B. Hartmann and W. Zakzewski, Electrons on hexagonal lattices and applications to nanotubes,, \emph{Phys. Rev.}, 68 (2003), 1. Google Scholar

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R. W. Hasse, A general method for the solution of nonlinear soliton and kink Schrödinger equations,, \emph{Z. Phys.}, 37 (1980), 83. doi: 10.1007/BF01325508. Google Scholar

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L. Jeanjean and K. Tanaka, A remark on least energy solutions in $\mathbbR^N$,, \emph{Proc. Amer. Math. Soc.}, 131 (2003), 2399. Google Scholar

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A. M. Kosevich, B. A. Ivanov and A. S. Kovalev, Magnetic solitons,, \emph{Phys. Rep.}, 194 (1990), 117. Google Scholar

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

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E. W. Laedke, K. H. 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

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A. G. Litvak and A. M. Sergeev, One dimensional collapse of plasma waves,, \emph{JETP Lett.}, 27 (1978), 517. Google Scholar

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J. Q. Liu and Z. Q. Wang, Soliton solutions for quasilinear Schrödinger equations I,, \emph{Proc. Amer. Math. Soc.}, 131 (2002), 441. Google Scholar

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J. Q. Liu, Y. Q, Wang and Z. Q. 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

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J. Q. Liu, Y. Q. Wang and Z. Q. Wang, Solutions for quasilinear Schrödinger equations via the Nehari Method,, \emph{Commun. Partial Differ. Equ.}, 29 (2004), 879. doi: 10.1081/PDE-120037335. Google Scholar

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V. G. Makhankov and V. K. Fedyanin, Non-linear effects in quasi-one-dimensional models of condensed matter theory,, \emph{Phys. Rep.}, 104 (1984), 1. doi: 10.1016/0370-1573(84)90106-6. Google Scholar

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M. Poppenberg, K. Schmitt and Z. Q. 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

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G. R. W. Quispel and H. W. Capel, Equation of motion for the Heisenberg spin chain,, \emph{Physica A}, 110 (1982), 41. doi: 10.1016/0378-4371(82)90104-2. Google Scholar

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D. Ruiz and G. Siciliano, Existence of ground states for a modified nonlinear Schrödinger equation,, \emph{Nonlinearity}, 23 (2010), 1221. doi: 10.1088/0951-7715/23/5/011. Google Scholar

[24]

Y. T. Shen and X. K. Guo, The positive solutions of degenerate variational problem and degenerate elliptic equations,, \emph{Chin. J. Cont. Mathematics}, 14 (1993), 157. Google Scholar

[25]

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

[26]

E. A. B. Silva and G. F. Vieira, Quasilinear asymptotically periodic Schrödinger equations with critical growth,, \emph{Calc. Var. Partial Differential Equations}, 39 (2010), 1. doi: 10.1007/s00526-009-0299-1. Google Scholar

[27]

Y. J. Wang, Y. M. Zhang and Y. T. Shen, Multiple solutions for quasilinear Schrödinger equations involving critical exponent,, \emph{Appl. Math. Comp.}, 216 (2010), 849. doi: 10.1016/j.amc.2010.01.091. Google Scholar

[28]

Y. J. Wang and W. M, Zou, Bound states to critical quasilinear Schrödinger equations,, \emph{Nonl. Diff. Equa. Appl.}, 19 (2012), 19. Google Scholar

[29]

J. Yang, Y. J. Wang and A. A. Abdelgadir, Soliton solutions for quasilinear Schrödinger equations,, \emph{J. Math. Phys}, 54 (2013). Google Scholar

show all references

References:
[1]

S. Adachia and T. Watanable, G-invariant positive solutions for a quasilinear Schrödinger equation,, \emph{Adv. Diff. Eqns.}, 16 (2011), 289. Google Scholar

[2]

S. Adachia and T. Watanabeb, Uniqueness of the ground state solutions of quasilinear Schrödinger equations,, \emph{Nonl. Anal. TMA.}, 75 (2012), 819. Google Scholar

[3]

A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications,, \emph{J. Funct. Anal.}, 14 (1973), 349. Google Scholar

[4]

H. Berestycki and P. L. Lions, Nonlinear scalar field equations I,, \emph{Arch. Rational Mech. Anal.}, 82 (1983), 313. doi: 10.1007/BF00250555. Google Scholar

[5]

H. Brézis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponent,, \emph{Comm. Pure Appl. Math.}, 36 (1983), 437. doi: 10.1002/cpa.3160360405. Google Scholar

[6]

L. Brizhik, A. Eremko, B. Piette and W. J. Zakrzewski, Static solutions of a D-dimensional modified nonlinear Schrödinger equation,, \emph{Nonlinearity}, 16 (2003), 1481. doi: 10.1088/0951-7715/16/4/317. Google Scholar

[7]

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

[8]

A. De Bouard, N. Hayashi and J. C. 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

[9]

J. M. do Ó, O. Miyagaki and H. Olimpio, 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

[10]

B. Hartmann and W. Zakzewski, Electrons on hexagonal lattices and applications to nanotubes,, \emph{Phys. Rev.}, 68 (2003), 1. Google Scholar

[11]

R. W. Hasse, A general method for the solution of nonlinear soliton and kink Schrödinger equations,, \emph{Z. Phys.}, 37 (1980), 83. doi: 10.1007/BF01325508. Google Scholar

[12]

L. Jeanjean and K. Tanaka, A remark on least energy solutions in $\mathbbR^N$,, \emph{Proc. Amer. Math. Soc.}, 131 (2003), 2399. Google Scholar

[13]

A. M. Kosevich, B. A. Ivanov and A. S. Kovalev, Magnetic solitons,, \emph{Phys. Rep.}, 194 (1990), 117. Google Scholar

[14]

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

[15]

E. W. Laedke, K. H. 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

[16]

A. G. Litvak and A. M. Sergeev, One dimensional collapse of plasma waves,, \emph{JETP Lett.}, 27 (1978), 517. Google Scholar

[17]

J. Q. Liu and Z. Q. Wang, Soliton solutions for quasilinear Schrödinger equations I,, \emph{Proc. Amer. Math. Soc.}, 131 (2002), 441. Google Scholar

[18]

J. Q. Liu, Y. Q, Wang and Z. Q. 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

[19]

J. Q. Liu, Y. Q. Wang and Z. Q. Wang, Solutions for quasilinear Schrödinger equations via the Nehari Method,, \emph{Commun. Partial Differ. Equ.}, 29 (2004), 879. doi: 10.1081/PDE-120037335. Google Scholar

[20]

V. G. Makhankov and V. K. Fedyanin, Non-linear effects in quasi-one-dimensional models of condensed matter theory,, \emph{Phys. Rep.}, 104 (1984), 1. doi: 10.1016/0370-1573(84)90106-6. Google Scholar

[21]

M. Poppenberg, K. Schmitt and Z. Q. 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

[22]

G. R. W. Quispel and H. W. Capel, Equation of motion for the Heisenberg spin chain,, \emph{Physica A}, 110 (1982), 41. doi: 10.1016/0378-4371(82)90104-2. Google Scholar

[23]

D. Ruiz and G. Siciliano, Existence of ground states for a modified nonlinear Schrödinger equation,, \emph{Nonlinearity}, 23 (2010), 1221. doi: 10.1088/0951-7715/23/5/011. Google Scholar

[24]

Y. T. Shen and X. K. Guo, The positive solutions of degenerate variational problem and degenerate elliptic equations,, \emph{Chin. J. Cont. Mathematics}, 14 (1993), 157. Google Scholar

[25]

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

[26]

E. A. B. Silva and G. F. Vieira, Quasilinear asymptotically periodic Schrödinger equations with critical growth,, \emph{Calc. Var. Partial Differential Equations}, 39 (2010), 1. doi: 10.1007/s00526-009-0299-1. Google Scholar

[27]

Y. J. Wang, Y. M. Zhang and Y. T. Shen, Multiple solutions for quasilinear Schrödinger equations involving critical exponent,, \emph{Appl. Math. Comp.}, 216 (2010), 849. doi: 10.1016/j.amc.2010.01.091. Google Scholar

[28]

Y. J. Wang and W. M, Zou, Bound states to critical quasilinear Schrödinger equations,, \emph{Nonl. Diff. Equa. Appl.}, 19 (2012), 19. Google Scholar

[29]

J. Yang, Y. J. Wang and A. A. Abdelgadir, Soliton solutions for quasilinear Schrödinger equations,, \emph{J. Math. Phys}, 54 (2013). Google Scholar

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