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November  2014, 13(6): 2273-2287. doi: 10.3934/cpaa.2014.13.2273

Positive solutions for quasilinear Schrödinger equations with critical growth and potential vanishing at infinity

Yinbin Deng 1, and  Wei Shuai 2,

1. 

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

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

Received  August 2013 Revised  May 2014 Published  July 2014

This paper is concerned with the existence of positive solutions for a class of quasilinear Schrödinger equations in $R^N$ with critical growth and potential vanishing at infinity. By using a change of variables, the quasilinear equations are reduced to semilinear one. Since the potential vanish at infinity, the associated functionals are still not well defined in the usual Sobolev space. So we have to work in the weighted Sobolev spaces, by Hardy-type inequality, then the functionals are well defined in the weighted Sobolev space and satisfy the geometric conditions of the Mountain Pass Theorem. Using this fact, we obtain a Cerami sequence converging weakly to a solution $v$. In the proof that $v$ is nontrivial, the main tool is classical arguments used by H. Brezis and L. Nirenberg in [13].
Keywords: critical growth, positive solution., Quasilinear Schrödinger equations, potential vanishing, weighted Sobolev space.
Mathematics Subject Classification: Primary: 35J20, 35J62; Secondary: 35Q5.
Citation: Yinbin Deng, Wei Shuai. Positive solutions for quasilinear Schrödinger equations with critical growth and potential vanishing at infinity. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2273-2287. doi: 10.3934/cpaa.2014.13.2273
References:
[1]

Claudianor O. Alves and Marco A. S. Souto, Existence of solutions for a class of nonlinear Schrödinger equations with potentials vanishing at infinitly,, \emph{J. Differential Equations, 254 (2013), 1977.  doi: 10.1016/j.jde.2012.11.013.  Google Scholar

[2]

A. Ambrosetti, V. Felli and A. Malchiodi, Ground states of nonlinear Schrödinger equations with potentials vanishing at infinitly,, \emph{J. Eur. Math. Soc.}, 7 (2005), 117.  doi: 10.4171/JEMS/24.  Google Scholar

[3]

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

[4]

A. Ambrosetti and Z-Q. Wang, Nonlinear Schrödinger equations with vanishing and decaying potentials,, \emph{Differential Integral Equations, 18 (2005), 1321.   Google Scholar

[5]

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

[6]

Denis Bonheure and Jean Van Schaftingen, Groundstates for the nonlinear Schrödinger equation with potentials vanishing at infinitly,, \emph{Ann. Mat. Pura Appl., 189 (2010), 273.  doi: 10.1007/s10231-009-0109-6.  Google Scholar

[7]

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

[8]

F. G. Bass and N. N. Nasanov, Nonlinear electromagnetic-spin waves,, \emph{Phys. Rep., 189 (1990), 165.  doi: 10.1016/0370-1573(90)90093-H.  Google Scholar

[9]

H. Berestycki and P. L. Lions, Nonlinear scalar field equations, I: Existence of a ground state,, \emph{Arch. Ration. Mech. Anal., 82 (1983), 313.  doi: 10.1007/BF00250555.  Google Scholar

[10]

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

[11]

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.  doi: 10.1063/1.870756.  Google Scholar

[12]

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

[13]

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

[14]

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

[15]

L. Brüll and H. Lange, Solitary waves for quasilinear Schrödinger equations,, \emph{Exposition. Math., 4 (1986), 279.   Google Scholar

[16]

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

[17]

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

[18]

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

[19]

S. Cuccagna, On instability of excited states of the nonloinear Schrödinger equation,, \emph{Physica D, 238 (2009), 38.  doi: 10.1016/j.physd.2008.08.010.  Google Scholar

[20]

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

[21]

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

[22]

Y. Deng, S. Peng and J. Wang, Nodal soliton solutions for quasilinear Schrödinger equations with critical exponent,, \emph{J. Math. Phys., 54 (2013), 1.  doi: 10.1063/1.4774153.  Google Scholar

[23]

A. Floer and A. Weinstein, Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential,, \emph{J. Funct. Anal., 69 (1986), 397.  doi: 10.1016/0022-1236(86)90096-0.  Google Scholar

[24]

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

[25]

A. M. Kosevich, B. A. Ivanov and A. S. Kovalev, Magnetic solitons,, \emph{\it Phys. Rep., 194 (1990), 117.  doi: 10.1016/0370-1573(90)90130-T.  Google Scholar

[26]

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

[27]

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

[28]

H. Lange, M. Poppenberg and H. Teismann, Nash-Moser methods for the solution of quasilinear Schrödinger equations,, \emph{Comm. Partial Differential Equations, 24 (1999), 1399.  doi: 10.1080/03605309908821469.  Google Scholar

[29]

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

[30]

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

[31]

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

[32]

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

[33]

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

[34]

S. Lorca and P. Ubilla, Symmetric and nonsymmetric solutions for an elliptic equation on $R^N$,, \emph{Nonlinear Anal., 58 (2004), 961.  doi: 10.1016/j.na.2004.03.034.  Google Scholar

[35]

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

[36]

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

[37]

M. Musso, F. Pacard and J. Wei, Finite-energy sign-changing solutions with dihedral symmetry for the stationary nonlinear Schrödinger equation,, \emph{J. Eur. Math. Soc.}, 14 (2012), 1923.  doi: 10.4171/JEMS/351.  Google Scholar

[38]

B. Opic and A. Kufner, Hardy-Type Inequalities,, Pitman Res. Notes Math. Ser. vol. 219. Longman Scientific and Technical. Harlow, (1990).   Google Scholar

[39]

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

[40]

P. Pucci and J. Serrin, A general variational identity,, \emph{Indiana Univ. Math. J., 35 (1986), 681.  doi: 10.1512/iumj.1986.35.35036.  Google Scholar

[41]

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

[42]

P. H. Rabinowitz, On a class of nonlinear Schrödinger equations,, \emph{Z. Angew. Math. Phys., 43 (1992), 270.  doi: 10.1007/BF00946631.  Google Scholar

[43]

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

[44]

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

show all references

References:
[1]

Claudianor O. Alves and Marco A. S. Souto, Existence of solutions for a class of nonlinear Schrödinger equations with potentials vanishing at infinitly,, \emph{J. Differential Equations, 254 (2013), 1977.  doi: 10.1016/j.jde.2012.11.013.  Google Scholar

[2]

A. Ambrosetti, V. Felli and A. Malchiodi, Ground states of nonlinear Schrödinger equations with potentials vanishing at infinitly,, \emph{J. Eur. Math. Soc.}, 7 (2005), 117.  doi: 10.4171/JEMS/24.  Google Scholar

[3]

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

[4]

A. Ambrosetti and Z-Q. Wang, Nonlinear Schrödinger equations with vanishing and decaying potentials,, \emph{Differential Integral Equations, 18 (2005), 1321.   Google Scholar

[5]

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

[6]

Denis Bonheure and Jean Van Schaftingen, Groundstates for the nonlinear Schrödinger equation with potentials vanishing at infinitly,, \emph{Ann. Mat. Pura Appl., 189 (2010), 273.  doi: 10.1007/s10231-009-0109-6.  Google Scholar

[7]

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

[8]

F. G. Bass and N. N. Nasanov, Nonlinear electromagnetic-spin waves,, \emph{Phys. Rep., 189 (1990), 165.  doi: 10.1016/0370-1573(90)90093-H.  Google Scholar

[9]

H. Berestycki and P. L. Lions, Nonlinear scalar field equations, I: Existence of a ground state,, \emph{Arch. Ration. Mech. Anal., 82 (1983), 313.  doi: 10.1007/BF00250555.  Google Scholar

[10]

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

[11]

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.  doi: 10.1063/1.870756.  Google Scholar

[12]

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

[13]

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

[14]

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

[15]

L. Brüll and H. Lange, Solitary waves for quasilinear Schrödinger equations,, \emph{Exposition. Math., 4 (1986), 279.   Google Scholar

[16]

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

[17]

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

[18]

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

[19]

S. Cuccagna, On instability of excited states of the nonloinear Schrödinger equation,, \emph{Physica D, 238 (2009), 38.  doi: 10.1016/j.physd.2008.08.010.  Google Scholar

[20]

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

[21]

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

[22]

Y. Deng, S. Peng and J. Wang, Nodal soliton solutions for quasilinear Schrödinger equations with critical exponent,, \emph{J. Math. Phys., 54 (2013), 1.  doi: 10.1063/1.4774153.  Google Scholar

[23]

A. Floer and A. Weinstein, Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential,, \emph{J. Funct. Anal., 69 (1986), 397.  doi: 10.1016/0022-1236(86)90096-0.  Google Scholar

[24]

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

[25]

A. M. Kosevich, B. A. Ivanov and A. S. Kovalev, Magnetic solitons,, \emph{\it Phys. Rep., 194 (1990), 117.  doi: 10.1016/0370-1573(90)90130-T.  Google Scholar

[26]

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

[27]

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

[28]

H. Lange, M. Poppenberg and H. Teismann, Nash-Moser methods for the solution of quasilinear Schrödinger equations,, \emph{Comm. Partial Differential Equations, 24 (1999), 1399.  doi: 10.1080/03605309908821469.  Google Scholar

[29]

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

[30]

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

[31]

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

[32]

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

[33]

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

[34]

S. Lorca and P. Ubilla, Symmetric and nonsymmetric solutions for an elliptic equation on $R^N$,, \emph{Nonlinear Anal., 58 (2004), 961.  doi: 10.1016/j.na.2004.03.034.  Google Scholar

[35]

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

[36]

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

[37]

M. Musso, F. Pacard and J. Wei, Finite-energy sign-changing solutions with dihedral symmetry for the stationary nonlinear Schrödinger equation,, \emph{J. Eur. Math. Soc.}, 14 (2012), 1923.  doi: 10.4171/JEMS/351.  Google Scholar

[38]

B. Opic and A. Kufner, Hardy-Type Inequalities,, Pitman Res. Notes Math. Ser. vol. 219. Longman Scientific and Technical. Harlow, (1990).   Google Scholar

[39]

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

[40]

P. Pucci and J. Serrin, A general variational identity,, \emph{Indiana Univ. Math. J., 35 (1986), 681.  doi: 10.1512/iumj.1986.35.35036.  Google Scholar

[41]

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

[42]

P. H. Rabinowitz, On a class of nonlinear Schrödinger equations,, \emph{Z. Angew. Math. Phys., 43 (1992), 270.  doi: 10.1007/BF00946631.  Google Scholar

[43]

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

[44]

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

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