# American Institute of Mathematical Sciences

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 . 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. Ambrosetti, V. 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. Ambrosetti, A. 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. Laedke, K. 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. Lange, B. 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. Liu, Y. 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. Liu, Z. 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$, J. Differ. Equ., 229 (2006), 570-587. doi: 10.1016/j.jde.2006.07.001. Google Scholar [21] A. Moameni, On the existence of standing wave solutions to quasilinear Schrödinger equations, Nonlinearity, 19 (2006), 937-957. doi: 10.1088/0951-7715/19/4/009. Google Scholar [22] V. Moroz and J. Van Schaftingen, Semiclassical stationary states for nonlinear Schrödinger equations with fastdecaying potentials, Calc. Var., 37 (2010), 1-27. doi: 10.1007/s00526-009-0249-y. Google Scholar [23] V. G. Makhankov and V. K. Fedanin, Nonlinear effects in quasi-one-dimensional models of condensed matter theory, Phys. Rep., 104 (1984), 1-86. doi: 10.1016/0370-1573(84)90106-6. Google Scholar [24] B. Opic and A. Kufner, Hardy-type Inequalities, Pitman Res., Notes in Math. Ser. 219, Longman Sci. Tech., Harlow, 1990. Google Scholar [25] B. Ritchie, Relativistic self-focusing and channel formation in laser-plasma interactions, Phys. Rev. E, 50 (1994), 687-689. Google Scholar [26] Y. T. Shen and Y. J. Wang, Soliton solutions for generalized quasilinear Schrödinger equations, Nonlinear Anal. TMA, 80 (2013), 194-201. doi: 10.1016/j.na.2012.10.005. Google Scholar [27] J. B. Su, Z. Q. Wang and M. Willem, Nonlinear Schrödinger equations with unbounded and decaying radial potentials, Commun. Cont. Math., 9 (2007), 571-583. doi: 10.1142/S021919970700254X. Google Scholar [28] Y. J. Wang, J. Yang and Y. M. Zhang, Quasilinear elliptic equations involving the$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. Ambrosetti, V. 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. Ambrosetti, A. 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. Laedke, K. 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. Lange, B. 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. Liu, Y. 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. Liu, Z. 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$, J. Differ. Equ., 229 (2006), 570-587. doi: 10.1016/j.jde.2006.07.001. Google Scholar [21] A. Moameni, On the existence of standing wave solutions to quasilinear Schrödinger equations, Nonlinearity, 19 (2006), 937-957. doi: 10.1088/0951-7715/19/4/009. Google Scholar [22] V. Moroz and J. Van Schaftingen, Semiclassical stationary states for nonlinear Schrödinger equations with fastdecaying potentials, Calc. Var., 37 (2010), 1-27. doi: 10.1007/s00526-009-0249-y. Google Scholar [23] V. G. Makhankov and V. K. Fedanin, Nonlinear effects in quasi-one-dimensional models of condensed matter theory, Phys. Rep., 104 (1984), 1-86. doi: 10.1016/0370-1573(84)90106-6. Google Scholar [24] B. Opic and A. Kufner, Hardy-type Inequalities, Pitman Res., Notes in Math. Ser. 219, Longman Sci. Tech., Harlow, 1990. Google Scholar [25] B. Ritchie, Relativistic self-focusing and channel formation in laser-plasma interactions, Phys. Rev. E, 50 (1994), 687-689. Google Scholar [26] Y. T. Shen and Y. J. Wang, Soliton solutions for generalized quasilinear Schrödinger equations, Nonlinear Anal. TMA, 80 (2013), 194-201. doi: 10.1016/j.na.2012.10.005. Google Scholar [27] J. B. Su, Z. Q. Wang and M. Willem, Nonlinear Schrödinger equations with unbounded and decaying radial potentials, Commun. Cont. Math., 9 (2007), 571-583. doi: 10.1142/S021919970700254X. Google Scholar [28] Y. J. Wang, J. Yang and Y. M. Zhang, Quasilinear elliptic equations involving the$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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