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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

 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].
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. [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. [3] A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications,, \emph{J. Func. Anal., 14 (1973), 349. [4] A. Ambrosetti and Z-Q. Wang, Nonlinear Schrödinger equations with vanishing and decaying potentials,, \emph{Differential Integral Equations, 18 (2005), 1321. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [15] L. Brüll and H. Lange, Solitary waves for quasilinear Schrödinger equations,, \emph{Exposition. Math., 4 (1986), 279. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [26] S. Kurihara, Large-amplitude quasi-solitons in superfluid films,, \emph{J. Phys. Soc. Japan, 50 (1981), 3262. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [38] B. Opic and A. Kufner, Hardy-Type Inequalities,, Pitman Res. Notes Math. Ser. vol. 219. Longman Scientific and Technical. Harlow, (1990). [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. [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. [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. [42] P. H. Rabinowitz, On a class of nonlinear Schrödinger equations,, \emph{Z. Angew. Math. Phys., 43 (1992), 270. doi: 10.1007/BF00946631. [43] B. Ritchie, Relativistic self-focusing and channel formation in laser-plasma interactions,, \emph{Phys. Rev. E, 50 (1994), 687. [44] W. A. Strauss, Existence of solitary waves in higher dimensions,, \emph{Comm. Math. Phys., 55 (1977), 149.

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. [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. [3] A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications,, \emph{J. Func. Anal., 14 (1973), 349. [4] A. Ambrosetti and Z-Q. Wang, Nonlinear Schrödinger equations with vanishing and decaying potentials,, \emph{Differential Integral Equations, 18 (2005), 1321. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [15] L. Brüll and H. Lange, Solitary waves for quasilinear Schrödinger equations,, \emph{Exposition. Math., 4 (1986), 279. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [26] S. Kurihara, Large-amplitude quasi-solitons in superfluid films,, \emph{J. Phys. Soc. Japan, 50 (1981), 3262. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [38] B. Opic and A. Kufner, Hardy-Type Inequalities,, Pitman Res. Notes Math. Ser. vol. 219. Longman Scientific and Technical. Harlow, (1990). [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. [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. [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. [42] P. H. Rabinowitz, On a class of nonlinear Schrödinger equations,, \emph{Z. Angew. Math. Phys., 43 (1992), 270. doi: 10.1007/BF00946631. [43] B. Ritchie, Relativistic self-focusing and channel formation in laser-plasma interactions,, \emph{Phys. Rev. E, 50 (1994), 687. [44] W. A. Strauss, Existence of solitary waves in higher dimensions,, \emph{Comm. Math. Phys., 55 (1977), 149.
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