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

Zhouxin Li; Yimin Zhang

doi:10.3934/cpaa.2020298

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2021, Volume 20, Issue 2: 933-954. Doi: 10.3934/cpaa.2020298

This issue Previous Article The degenerate Monge-Ampère equations with the Neumann condition Next Article

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

Zhouxin Li^1, and
Yimin Zhang^2, ,

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
^* Corresponding author

Received: July 31, 2020

Revised: September 30, 2020

Early access: December 2020

Published: February 2021

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)

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Abstract

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.

Keywords:

Mathematics Subject Classification: Primary: 35J10, 35J20; Secondary: 35J25.

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[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.
[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.
[3]	A. Ambrosetti and Z. Q. Wang, Nonlinear Schrödinger equations with vanishing and decaying potentials, Differ. Integral Equ., 18 (2005), 1321-1332.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[12]	S. Kurihura, Large-amplitude quasi-solitions in superfluid films, J. Phys. Soc. Jpn, 50 (1981), 3262-3267. doi: 10.1143/JPSJ.50.3262.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[24]	B. Opic and A. Kufner, Hardy-type Inequalities, Pitman Res., Notes in Math. Ser. 219, Longman Sci. Tech., Harlow, 1990.
[25]	B. Ritchie, Relativistic self-focusing and channel formation in laser-plasma interactions, Phys. Rev. E, 50 (1994), 687-689.
[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.
[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.
[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.
[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.