February  2021, 20(2): 933-954. doi: 10.3934/cpaa.2020298

Ground states for a class of quasilinear Schrödinger equations with vanishing potentials

1. 

School of Mathematics and Statistics, Central South University, Changsha, 410083, China

2. 

Center for Mathematical Sciences, Wuhan University of Technology, Wuhan, 430070, China

* Corresponding author

Received  August 2020 Revised  October 2020 Published  December 2020

Fund Project: The first author is supported by Hunan Provincial Natural Science Foundation under grant number 2019JJ40355; The second author is supported by the Natural Science Foundation of China under grant number 11771127 and the Fundamental Research Funds for Central Universities (WUT: 2019IB009, 2020IB011, 2020IB019)

In this paper, we study a class of quasilinear Schrödinger equation of the form
$ -\varepsilon^2\Delta u+V(x)u-\varepsilon^2(\Delta(|u|^{2}))u = K(x)|u|^{q-2}u,\quad x\in{\mathbb{R}^N}, $
where
$ V $
,
$ K $
are smooth functions and
$ V $
may vanish at infinity,
$ 2<q<2(2^*) $
. We prove the existence of a positive ground state solution which possesses a unique local maximum and decays exponentially.
Citation: Zhouxin Li, Yimin Zhang. Ground states for a class of quasilinear Schrödinger equations with vanishing potentials. Communications on Pure & Applied Analysis, 2021, 20 (2) : 933-954. doi: 10.3934/cpaa.2020298
References:
[1]

A. AmbrosettiV. Felli and A. Malchiodi, Ground states of nonlinear Schrödinger equations with potentials vanishing at infinity, J. Eur. Math. Soc., 7 (2005), 117-144.  doi: 10.4171/JEMS/24.  Google Scholar

[2]

A. AmbrosettiA. Malchiodi and D. Ruiz, Bound states of nonlinear Schrödinger equations with potentials vanishing at infinitey, J. Anal. Math., 98 (2006), 317-348.  doi: 10.1007/BF02790279.  Google Scholar

[3]

A. Ambrosetti and Z. Q. Wang, Nonlinear Schrödinger equations with vanishing and decaying potentials, Differ. Integral Equ., 18 (2005), 1321-1332.   Google Scholar

[4]

J. F. L. Aires and M. A. S. Souto, Existence of solutions for a quasilinear Schrödinger equation with vanishing potentials, J. Math. Anal. Appl., 416 (2014), 924-946.  doi: 10.1016/j.jmaa.2014.03.018.  Google Scholar

[5]

D. Bonheure and J. Van Schaftingen, Nonlinear Schrödinger equations with potentials vanishing at infinity, C. R. Math. Acad. Sci. Paris, 342 (2006), 903-908.  doi: 10.1016/j.crma.2006.04.011.  Google Scholar

[6]

D. Bonheure and J. Van Schaftingen, Bound state solutions for a class of nonlinear Schrödinger equations, Rev. Mat. Iberoam., 24 (2008), 297-351.  doi: 10.4171/RMI/537.  Google Scholar

[7]

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

[8]

M. del Pino and P. L. Felmer, Local mountain passes for semilinear elliptic problems in unbounded domains, Calc. Var. Partial Differ. Equ., 4 (1996), 121-137.  doi: 10.1007/BF01189950.  Google Scholar

[9]

Y. B. Deng and W. Shuai, Positive solutions for quasilinear Schrödinger equations with critical growth and potential vanishing at infinity, Commun. Pure Appl. Anal., 13 (2014), 2273-2287.  doi: 10.3934/cpaa.2014.13.2273.  Google Scholar

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J. M. Do Ó and U. Severo, Quasilinear Schrödinger equations involving concave and convex nonlinearities, Commun. Pure Appl. Anal., 8 (2009), 621-644.  doi: 10.3934/cpaa.2009.8.621.  Google Scholar

[11]

R. W. Hasse, A general method for the solution of nonlinear soliton and kink Schrödinger equations, Z. Phys. B, 37 (1980), 83-87.  doi: 10.1007/BF01325508.  Google Scholar

[12]

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

[13]

E. W. LaedkeK. H. Spatschek and L. Stenflo, Evolution theorem for a class of perturbed envelope soliton solutions, J. Math. Phys., 24 (1983), 2764-2769.  doi: 10.1063/1.525675.  Google Scholar

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H. LangeB. Toomire and P. F. Zweifel, Time-dependent dissipation in nonlinear Schrödinger systems, J. Math. Phys., 36 (1995), 1274-1283.  doi: 10.1063/1.531120.  Google Scholar

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Z. X. Li, Positive solutions for a class of singular quasilinear Schrödinger equations with critical Sobolev exponent, J. Differ. Equ., 266 (2019), 7264-7290.  doi: 10.1016/j.jde.2018.11.030.  Google Scholar

[16]

Z. X. Li and Y. M. Zhang, Solutions for a class of quasilinear Schrödinger equations with critical Sobolev exponents, J. Math. Phys., 58 (2017), 021501. doi: 10.1063/1.4975009.  Google Scholar

[17]

J. Q. Liu and Z. Q. Wang, Soliton solutions for quasilinear Schrödinger equations, Ⅰ, Proc. Amer. Math. Soc., 131 (2003), 441-448.  doi: 10.1090/S0002-9939-02-06783-7.  Google Scholar

[18]

J. Q. LiuY. Q. Wang and Z. Q. Wang, Soliton solutions for quasilinear Schrödinger equations, Ⅱ, J. Differ. Equ., 187 (2003), 473-493.  doi: 10.1016/S0022-0396(02)00064-5.  Google Scholar

[19]

C. Y. LiuZ. P. Wang and H. S. Zhou, Asymptotically linear Schrödinger equaiton with potential vanishing at infinity, J. Differ. Equ., 245 (2008), 201-222.  doi: 10.1016/j.jde.2008.01.006.  Google Scholar

[20]

A. Moameni, Existence of soliton solutions for a quasilinear Schrödinger equation involving critical exponent in ${\mathbb{R}}^N$, J. Differ. Equ., 229 (2006), 570-587.  doi: 10.1016/j.jde.2006.07.001.  Google Scholar

[21]

A. Moameni, On the existence of standing wave solutions to quasilinear Schrödinger equations, Nonlinearity, 19 (2006), 937-957.  doi: 10.1088/0951-7715/19/4/009.  Google Scholar

[22]

V. Moroz and J. Van Schaftingen, Semiclassical stationary states for nonlinear Schrödinger equations with fastdecaying potentials, Calc. Var., 37 (2010), 1-27.  doi: 10.1007/s00526-009-0249-y.  Google Scholar

[23]

V. G. Makhankov and V. K. Fedanin, Nonlinear effects in quasi-one-dimensional models of condensed matter theory, Phys. Rep., 104 (1984), 1-86.  doi: 10.1016/0370-1573(84)90106-6.  Google Scholar

[24]

B. Opic and A. Kufner, Hardy-type Inequalities, Pitman Res., Notes in Math. Ser. 219, Longman Sci. Tech., Harlow, 1990.  Google Scholar

[25]

B. Ritchie, Relativistic self-focusing and channel formation in laser-plasma interactions, Phys. Rev. E, 50 (1994), 687-689.   Google Scholar

[26]

Y. T. Shen and Y. J. Wang, Soliton solutions for generalized quasilinear Schrödinger equations, Nonlinear Anal. TMA, 80 (2013), 194-201.  doi: 10.1016/j.na.2012.10.005.  Google Scholar

[27]

J. B. SuZ. Q. Wang and M. Willem, Nonlinear Schrödinger equations with unbounded and decaying radial potentials, Commun. Cont. Math., 9 (2007), 571-583.  doi: 10.1142/S021919970700254X.  Google Scholar

[28]

Y. J. WangJ. Yang and Y. M. Zhang, Quasilinear elliptic equations involving the $N$-Laplacian with critical exponential growth in ${\mathbb{R}}^N$, Nonlinear Anal. TMA, 71 (2009), 6157-6169.  doi: 10.1016/j.na.2009.06.006.  Google Scholar

[29]

Z. P. Wang and H. S. Zhou, Ground state for nonlinear Schrödinger equation with sign-changing and vanishing potential, J. Math. Phys., 52 (2011), 113704. doi: 10.1063/1.3663434.  Google Scholar

show all references

References:
[1]

A. AmbrosettiV. Felli and A. Malchiodi, Ground states of nonlinear Schrödinger equations with potentials vanishing at infinity, J. Eur. Math. Soc., 7 (2005), 117-144.  doi: 10.4171/JEMS/24.  Google Scholar

[2]

A. AmbrosettiA. Malchiodi and D. Ruiz, Bound states of nonlinear Schrödinger equations with potentials vanishing at infinitey, J. Anal. Math., 98 (2006), 317-348.  doi: 10.1007/BF02790279.  Google Scholar

[3]

A. Ambrosetti and Z. Q. Wang, Nonlinear Schrödinger equations with vanishing and decaying potentials, Differ. Integral Equ., 18 (2005), 1321-1332.   Google Scholar

[4]

J. F. L. Aires and M. A. S. Souto, Existence of solutions for a quasilinear Schrödinger equation with vanishing potentials, J. Math. Anal. Appl., 416 (2014), 924-946.  doi: 10.1016/j.jmaa.2014.03.018.  Google Scholar

[5]

D. Bonheure and J. Van Schaftingen, Nonlinear Schrödinger equations with potentials vanishing at infinity, C. R. Math. Acad. Sci. Paris, 342 (2006), 903-908.  doi: 10.1016/j.crma.2006.04.011.  Google Scholar

[6]

D. Bonheure and J. Van Schaftingen, Bound state solutions for a class of nonlinear Schrödinger equations, Rev. Mat. Iberoam., 24 (2008), 297-351.  doi: 10.4171/RMI/537.  Google Scholar

[7]

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

[8]

M. del Pino and P. L. Felmer, Local mountain passes for semilinear elliptic problems in unbounded domains, Calc. Var. Partial Differ. Equ., 4 (1996), 121-137.  doi: 10.1007/BF01189950.  Google Scholar

[9]

Y. B. Deng and W. Shuai, Positive solutions for quasilinear Schrödinger equations with critical growth and potential vanishing at infinity, Commun. Pure Appl. Anal., 13 (2014), 2273-2287.  doi: 10.3934/cpaa.2014.13.2273.  Google Scholar

[10]

J. M. Do Ó and U. Severo, Quasilinear Schrödinger equations involving concave and convex nonlinearities, Commun. Pure Appl. Anal., 8 (2009), 621-644.  doi: 10.3934/cpaa.2009.8.621.  Google Scholar

[11]

R. W. Hasse, A general method for the solution of nonlinear soliton and kink Schrödinger equations, Z. Phys. B, 37 (1980), 83-87.  doi: 10.1007/BF01325508.  Google Scholar

[12]

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

[13]

E. W. LaedkeK. H. Spatschek and L. Stenflo, Evolution theorem for a class of perturbed envelope soliton solutions, J. Math. Phys., 24 (1983), 2764-2769.  doi: 10.1063/1.525675.  Google Scholar

[14]

H. LangeB. Toomire and P. F. Zweifel, Time-dependent dissipation in nonlinear Schrödinger systems, J. Math. Phys., 36 (1995), 1274-1283.  doi: 10.1063/1.531120.  Google Scholar

[15]

Z. X. Li, Positive solutions for a class of singular quasilinear Schrödinger equations with critical Sobolev exponent, J. Differ. Equ., 266 (2019), 7264-7290.  doi: 10.1016/j.jde.2018.11.030.  Google Scholar

[16]

Z. X. Li and Y. M. Zhang, Solutions for a class of quasilinear Schrödinger equations with critical Sobolev exponents, J. Math. Phys., 58 (2017), 021501. doi: 10.1063/1.4975009.  Google Scholar

[17]

J. Q. Liu and Z. Q. Wang, Soliton solutions for quasilinear Schrödinger equations, Ⅰ, Proc. Amer. Math. Soc., 131 (2003), 441-448.  doi: 10.1090/S0002-9939-02-06783-7.  Google Scholar

[18]

J. Q. LiuY. Q. Wang and Z. Q. Wang, Soliton solutions for quasilinear Schrödinger equations, Ⅱ, J. Differ. Equ., 187 (2003), 473-493.  doi: 10.1016/S0022-0396(02)00064-5.  Google Scholar

[19]

C. Y. LiuZ. P. Wang and H. S. Zhou, Asymptotically linear Schrödinger equaiton with potential vanishing at infinity, J. Differ. Equ., 245 (2008), 201-222.  doi: 10.1016/j.jde.2008.01.006.  Google Scholar

[20]

A. Moameni, Existence of soliton solutions for a quasilinear Schrödinger equation involving critical exponent in ${\mathbb{R}}^N$, J. Differ. Equ., 229 (2006), 570-587.  doi: 10.1016/j.jde.2006.07.001.  Google Scholar

[21]

A. Moameni, On the existence of standing wave solutions to quasilinear Schrödinger equations, Nonlinearity, 19 (2006), 937-957.  doi: 10.1088/0951-7715/19/4/009.  Google Scholar

[22]

V. Moroz and J. Van Schaftingen, Semiclassical stationary states for nonlinear Schrödinger equations with fastdecaying potentials, Calc. Var., 37 (2010), 1-27.  doi: 10.1007/s00526-009-0249-y.  Google Scholar

[23]

V. G. Makhankov and V. K. Fedanin, Nonlinear effects in quasi-one-dimensional models of condensed matter theory, Phys. Rep., 104 (1984), 1-86.  doi: 10.1016/0370-1573(84)90106-6.  Google Scholar

[24]

B. Opic and A. Kufner, Hardy-type Inequalities, Pitman Res., Notes in Math. Ser. 219, Longman Sci. Tech., Harlow, 1990.  Google Scholar

[25]

B. Ritchie, Relativistic self-focusing and channel formation in laser-plasma interactions, Phys. Rev. E, 50 (1994), 687-689.   Google Scholar

[26]

Y. T. Shen and Y. J. Wang, Soliton solutions for generalized quasilinear Schrödinger equations, Nonlinear Anal. TMA, 80 (2013), 194-201.  doi: 10.1016/j.na.2012.10.005.  Google Scholar

[27]

J. B. SuZ. Q. Wang and M. Willem, Nonlinear Schrödinger equations with unbounded and decaying radial potentials, Commun. Cont. Math., 9 (2007), 571-583.  doi: 10.1142/S021919970700254X.  Google Scholar

[28]

Y. J. WangJ. Yang and Y. M. Zhang, Quasilinear elliptic equations involving the $N$-Laplacian with critical exponential growth in ${\mathbb{R}}^N$, Nonlinear Anal. TMA, 71 (2009), 6157-6169.  doi: 10.1016/j.na.2009.06.006.  Google Scholar

[29]

Z. P. Wang and H. S. Zhou, Ground state for nonlinear Schrödinger equation with sign-changing and vanishing potential, J. Math. Phys., 52 (2011), 113704. doi: 10.1063/1.3663434.  Google Scholar

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