doi: 10.3934/dcdsb.2021214
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Positive solutions for critical quasilinear Schrödinger equations with potentials vanishing at infinity

1. 

College of Mathematics and Statistics, Qujing Normal University, Qujing 655011, Yunnan, China

2. 

Department of Mathematics, Soochow University, Suzhou 215006, Jiangsu, China

* Corresponding author: Guofa Li

Received  April 2021 Revised  June 2021 Early access August 2021

Fund Project: This work was supported by the National Natural Science Foundation of China (Grant Nos. 11971339, 11901345), the Yunnan Local Colleges Applied Basic Research Projects (Grant No. 202001BA070001-032) and Scientific and Technological Innovation Team of Nonlinear Analysis and Algebra with Their Applications in Universities of Yunnan Province, China (Grant No. 2020CXTD25)

In this paper, we study the existence of positive solutions for the following quasilinear Schrödinger equations
$ \begin{equation*} -\triangle u+V(x)u+\frac{\kappa}{2}[\triangle|u|^{2}]u = \lambda K(x)h(u)+\mu|u|^{2^*-2}u, \quad x\in\mathbb{R}^{N}, \end{equation*} $
where
$ \kappa>0 $
,
$ \lambda>0, \mu>0, h\in C(\mathbb{R}, \mathbb{R}) $
is superlinear at infinity, the potentials
$ V(x) $
and
$ K(x) $
are vanishing at infinity. In order to discuss the existence of solutions we apply minimax techniques together with careful
$ L^{\infty} $
-estimates. For the subcritical case (
$ \mu = 0 $
) we can deal with large
$ \kappa>0 $
. For the critical case we treat that
$ \kappa>0 $
is small.
Citation: Guofa Li, Yisheng Huang. Positive solutions for critical quasilinear Schrödinger equations with potentials vanishing at infinity. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021214
References:
[1]

J. F. L. Aires and M. A. S. Souto, Equation with positive coefficient in the quasilinear term and vanishing potential, Topol. Methods Nonlinear Anal., 46 (2015), 813-833. 

[2]

C. O. Alves and M. A. S. Souto, Existence of solutions for a class of nonlinear Schrödinger equations with potential vanishing at infinity, J. Differential Equations, 254 (2013), 1977-1991.  doi: 10.1016/j.jde.2012.11.013.

[3]

C. O. AlvesY. Wang and Y. Shen, Soliton solutions for a class of quasilinear Schrödinger equations with a parameter, J. Differential Equations, 259 (2015), 318-343.  doi: 10.1016/j.jde.2015.02.030.

[4]

H. Berestycki and P.-L. Lions, Nonlinear scalar field equations Ⅰ Existence of a ground state, Arch. Ration. Mech. Anal., 82 (1983), 313-346.  doi: 10.1007/BF00250555.

[5]

H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponent, Commun. Pure Appl. Math., 36 (1983), 437-477.  doi: 10.1002/cpa.3160360405.

[6]

L. BrizhikA. EremkoB. Piette and W. J. Zakrzewski, Electron self-trapping in a discrete two-dimensional lattice, Physica D, 159 (2001), 71-90.  doi: 10.1016/S0167-2789(01)00332-3.

[7]

L. BrizhikA. EremkoB. Piette and W. J. Zakrzewski, Static solutions of a D-dimensional modified nonlinear Schrödinger equation, Nonlinearity, 16 (2003), 1481-1497.  doi: 10.1088/0951-7715/16/4/317.

[8]

J. ChenX. Huang and B. Cheng, Positive solutions for a class of quasilinear Schrödinger equations with superlinear condition, Appl. Math. Lett., 87 (2019), 165-171.  doi: 10.1016/j.aml.2018.07.035.

[9]

J. Chen, X. Huang, B. Cheng and C. Zhu, Some results on standing wave solutions for a class of quasilinear Schrödinger equations, J. Math. Phys., 60 (2019), 091506, 55 pp. doi: 10.1063/1.5093720.

[10]

A. de BouardN. Hayashi and J.-C. Saut, Global existence of small solutions to a relativistic nonlinear Schrödinger equation, Comm. Math. Phys., 189 (1997), 73-105.  doi: 10.1007/s002200050191.

[11]

Y. Deng and W. Shuai, Non-trivial solutions for a semilinear biharmonic problem with critical growth and potential vanishing at infinity, Proc. Roy. Soc. Edinburgh Sect. A, 145 (2015), 281-299.  doi: 10.1017/S0308210513001170.

[12]

B. Hartmann and W. J. Zakrzewski, Electrons on hexagonal lattices and applications to nanotubes, Phys. Rev. B, 68 (2003), 184302. doi: 10.1103/PhysRevB.68.184302.

[13]

C. Huang and G. Jia, Existence of positive solutions for supercritical quasilinear Schrödinger elliptic equations, J. Mathe. Anal. Appl., 472 (2019), 705-727.  doi: 10.1016/j.jmaa.2018.11.048.

[14]

L. Jeanjean and K. Tanaka, A positive solution for a nonlinear Schrödinger equation on $\mathbb{R}^{N}$, Indiana Univ. Math. J., 54 (2005), 443-464.  doi: 10.1512/iumj.2005.54.2502.

[15]

S. Kurihura, Large-amplitude quasi-solitons in superfluid films, J. Phys. Soc. Japan, 50 (1981), 3262-3267.  doi: 10.1143/JPSJ.50.3262.

[16]

Q. Li and X. Wu, Existence, multiplicity, and concentration of solutions for generalized quasilinear Schödinger equations with critical growth, J. Math. Phys., 58 (2017), 041501. doi: 10.1063/1.4982035.

[17]

Z. Liang, J. Gao and A. Li, Existence of positive solutions for a class of quasilinear Schrödinger equations with local superlinear nonlinearities, J. Math. Anal. Appl., 484 (2020), 123732. doi: 10.1016/j.jmaa.2019.123732.

[18]

P. L. Lions, The concentration-compactness principle in the calculus of variation. The locally compact case, part Ⅰ and Ⅱ, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 109–145 and 223–283. doi: 10.1016/S0294-1449(16)30422-X.

[19]

U. B. Severo and G. M. de Carvalho, Quasilinear Schrödinger equations with a positive parameter and involving unbounded or decaying potentials, Appl. Anal., 100 (2021), 229-252.  doi: 10.1080/00036811.2019.1599106.

[20]

U. B. SeveroE. Gloss and E. D. da Silva, On a class of quasilinear Schrödinger equations with superlinear or asymptotically linear terms, J. Differential Equations, 263 (2017), 3550-3580.  doi: 10.1016/j.jde.2017.04.040.

[21]

Y. Shen and Y. Wang, Standing waves for a class of quasilinear Schrödinger equations, Complex Var. Elliptic Equ., 61 (2016), 817-842.  doi: 10.1080/17476933.2015.1119818.

[22]

Y. Wang, A class of quasilinear Schrödinger equations with critical or supercritical exponents, Compu. Math. Appl., 70 (2015), 562-572.  doi: 10.1016/j.camwa.2015.05.016.

[23]

Y. Wang and Z. Li, Existence of solutions to quasilinear Schrödinger equations involving critical Sobolev exponent, Taiwanese Journal of Math., 22 (2018), 401-420.  doi: 10.11650/tjm/8150.

[24]

M. Willem, Minimax Theorems, Birkhäuser Boston, Inc., Boston, MA, 1996. doi: 10.1007/978-1-4612-4146-1.

[25]

M. Yang, C. A. Santos and J. Zhou, Least action nodal solutions for a quasilinear defocusing Schrödinger equation with supercritical nonlinearity, Commun. Contemp. Math., 21 (2019), 1850026, 23 pp. doi: 10.1142/S0219199718500268.

[26]

W. Zou, Sign-Changing Critical Points Theory, Springer, New York, 2008.

show all references

References:
[1]

J. F. L. Aires and M. A. S. Souto, Equation with positive coefficient in the quasilinear term and vanishing potential, Topol. Methods Nonlinear Anal., 46 (2015), 813-833. 

[2]

C. O. Alves and M. A. S. Souto, Existence of solutions for a class of nonlinear Schrödinger equations with potential vanishing at infinity, J. Differential Equations, 254 (2013), 1977-1991.  doi: 10.1016/j.jde.2012.11.013.

[3]

C. O. AlvesY. Wang and Y. Shen, Soliton solutions for a class of quasilinear Schrödinger equations with a parameter, J. Differential Equations, 259 (2015), 318-343.  doi: 10.1016/j.jde.2015.02.030.

[4]

H. Berestycki and P.-L. Lions, Nonlinear scalar field equations Ⅰ Existence of a ground state, Arch. Ration. Mech. Anal., 82 (1983), 313-346.  doi: 10.1007/BF00250555.

[5]

H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponent, Commun. Pure Appl. Math., 36 (1983), 437-477.  doi: 10.1002/cpa.3160360405.

[6]

L. BrizhikA. EremkoB. Piette and W. J. Zakrzewski, Electron self-trapping in a discrete two-dimensional lattice, Physica D, 159 (2001), 71-90.  doi: 10.1016/S0167-2789(01)00332-3.

[7]

L. BrizhikA. EremkoB. Piette and W. J. Zakrzewski, Static solutions of a D-dimensional modified nonlinear Schrödinger equation, Nonlinearity, 16 (2003), 1481-1497.  doi: 10.1088/0951-7715/16/4/317.

[8]

J. ChenX. Huang and B. Cheng, Positive solutions for a class of quasilinear Schrödinger equations with superlinear condition, Appl. Math. Lett., 87 (2019), 165-171.  doi: 10.1016/j.aml.2018.07.035.

[9]

J. Chen, X. Huang, B. Cheng and C. Zhu, Some results on standing wave solutions for a class of quasilinear Schrödinger equations, J. Math. Phys., 60 (2019), 091506, 55 pp. doi: 10.1063/1.5093720.

[10]

A. de BouardN. Hayashi and J.-C. Saut, Global existence of small solutions to a relativistic nonlinear Schrödinger equation, Comm. Math. Phys., 189 (1997), 73-105.  doi: 10.1007/s002200050191.

[11]

Y. Deng and W. Shuai, Non-trivial solutions for a semilinear biharmonic problem with critical growth and potential vanishing at infinity, Proc. Roy. Soc. Edinburgh Sect. A, 145 (2015), 281-299.  doi: 10.1017/S0308210513001170.

[12]

B. Hartmann and W. J. Zakrzewski, Electrons on hexagonal lattices and applications to nanotubes, Phys. Rev. B, 68 (2003), 184302. doi: 10.1103/PhysRevB.68.184302.

[13]

C. Huang and G. Jia, Existence of positive solutions for supercritical quasilinear Schrödinger elliptic equations, J. Mathe. Anal. Appl., 472 (2019), 705-727.  doi: 10.1016/j.jmaa.2018.11.048.

[14]

L. Jeanjean and K. Tanaka, A positive solution for a nonlinear Schrödinger equation on $\mathbb{R}^{N}$, Indiana Univ. Math. J., 54 (2005), 443-464.  doi: 10.1512/iumj.2005.54.2502.

[15]

S. Kurihura, Large-amplitude quasi-solitons in superfluid films, J. Phys. Soc. Japan, 50 (1981), 3262-3267.  doi: 10.1143/JPSJ.50.3262.

[16]

Q. Li and X. Wu, Existence, multiplicity, and concentration of solutions for generalized quasilinear Schödinger equations with critical growth, J. Math. Phys., 58 (2017), 041501. doi: 10.1063/1.4982035.

[17]

Z. Liang, J. Gao and A. Li, Existence of positive solutions for a class of quasilinear Schrödinger equations with local superlinear nonlinearities, J. Math. Anal. Appl., 484 (2020), 123732. doi: 10.1016/j.jmaa.2019.123732.

[18]

P. L. Lions, The concentration-compactness principle in the calculus of variation. The locally compact case, part Ⅰ and Ⅱ, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 109–145 and 223–283. doi: 10.1016/S0294-1449(16)30422-X.

[19]

U. B. Severo and G. M. de Carvalho, Quasilinear Schrödinger equations with a positive parameter and involving unbounded or decaying potentials, Appl. Anal., 100 (2021), 229-252.  doi: 10.1080/00036811.2019.1599106.

[20]

U. B. SeveroE. Gloss and E. D. da Silva, On a class of quasilinear Schrödinger equations with superlinear or asymptotically linear terms, J. Differential Equations, 263 (2017), 3550-3580.  doi: 10.1016/j.jde.2017.04.040.

[21]

Y. Shen and Y. Wang, Standing waves for a class of quasilinear Schrödinger equations, Complex Var. Elliptic Equ., 61 (2016), 817-842.  doi: 10.1080/17476933.2015.1119818.

[22]

Y. Wang, A class of quasilinear Schrödinger equations with critical or supercritical exponents, Compu. Math. Appl., 70 (2015), 562-572.  doi: 10.1016/j.camwa.2015.05.016.

[23]

Y. Wang and Z. Li, Existence of solutions to quasilinear Schrödinger equations involving critical Sobolev exponent, Taiwanese Journal of Math., 22 (2018), 401-420.  doi: 10.11650/tjm/8150.

[24]

M. Willem, Minimax Theorems, Birkhäuser Boston, Inc., Boston, MA, 1996. doi: 10.1007/978-1-4612-4146-1.

[25]

M. Yang, C. A. Santos and J. Zhou, Least action nodal solutions for a quasilinear defocusing Schrödinger equation with supercritical nonlinearity, Commun. Contemp. Math., 21 (2019), 1850026, 23 pp. doi: 10.1142/S0219199718500268.

[26]

W. Zou, Sign-Changing Critical Points Theory, Springer, New York, 2008.

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