EXISTENCE AND CONCENTRATION BEHAVIOR OF GROUND STATE SOLUTIONS FOR A CLASS OF GENERALIZED QUASILINEAR SCHRÖDINGER EQUATIONS IN $\mathbb{R}^N$

doi:10.1007/s10473-020-0519-5

数学物理学报(英文版) ›› 2020, Vol. 40 ›› Issue (5): 1495-1524.doi: 10.1007/s10473-020-0519-5

EXISTENCE AND CONCENTRATION BEHAVIOR OF GROUND STATE SOLUTIONS FOR A CLASS OF GENERALIZED QUASILINEAR SCHRÖDINGER EQUATIONS IN $\mathbb{R}^N$

陈建华¹, 黄先玖¹, 程毕陶², 唐先华³

1. Department of Mathematics, Nanchang University, Nanchang 330031, China;
2. School of Mathematics and Statistics, Qujing Normal University, Qujing 655011, China;
3. School of Mathematics and Statistics, Central South University, Changsha 410083, China

收稿日期:2018-09-27 修回日期:2020-06-01 出版日期:2020-10-25 发布日期:2020-11-04
通讯作者: Xianjiu HUANG E-mail:xjhuangxwen@163.com
作者简介:Jianhua CHEN,E-mail:cjh19881129@163.com;Bitao CHENG,E-mail:chengbitao2006@126.com;Xianhua TANG,E-mail:tangxh@mail.csu.edu.cn
基金资助:
This work was supported by the National Natural Science Foundation of China (11661053, 11771198, 11901345, 11901276, 11961045 and 11971485), and partly by the Provincial Natural Science Foundation of Jiangxi, China (20161BAB201009 and 20181BAB201003), the Outstanding Youth Scientist Foundation Plan of Jiangxi (20171BCB23004), and the Yunnan Local Colleges Applied Basic Research Projects (2017FH001-011).

EXISTENCE AND CONCENTRATION BEHAVIOR OF GROUND STATE SOLUTIONS FOR A CLASS OF GENERALIZED QUASILINEAR SCHRÖDINGER EQUATIONS IN $\mathbb{R}^N$

Jianhua CHEN¹, Xianjiu HUANG¹, Bitao CHENG², Xianhua TANG³

1. Department of Mathematics, Nanchang University, Nanchang 330031, China;
2. School of Mathematics and Statistics, Qujing Normal University, Qujing 655011, China;
3. School of Mathematics and Statistics, Central South University, Changsha 410083, China

Received:2018-09-27 Revised:2020-06-01 Online:2020-10-25 Published:2020-11-04
Contact: Xianjiu HUANG E-mail:xjhuangxwen@163.com
Supported by:
This work was supported by the National Natural Science Foundation of China (11661053, 11771198, 11901345, 11901276, 11961045 and 11971485), and partly by the Provincial Natural Science Foundation of Jiangxi, China (20161BAB201009 and 20181BAB201003), the Outstanding Youth Scientist Foundation Plan of Jiangxi (20171BCB23004), and the Yunnan Local Colleges Applied Basic Research Projects (2017FH001-011).

摘要/Abstract

摘要： In this article, we study the generalized quasilinear Schrödinger equation \begin{equation*} -\text{div}(\varepsilon^2g^2(u)\nabla u)+\varepsilon^2g(u)g'(u)|\nabla u|^2+V(x)u=K(x)|u|^{p-2}u,\,\, x\in\mathbb{R}^N, \end{equation*} where $N\geq3$, $\varepsilon>0$, $4 < p < 22^*$, $g\in\mathcal{C}^1(\mathbb{R},\mathbb{R}^{+})$, $V\in \mathcal{C}(\mathbb{R}^N)\cap L^\infty(\mathbb{R}^N)$ has a positive global minimum, and $K\in \mathcal{C}(\mathbb{R}^N)\cap L^\infty(\mathbb{R}^N)$ has a positive global maximum. By using a change of variable, we obtain the existence and concentration behavior of ground state solutions for this problem and establish a phenomenon of exponential decay.

关键词: generalized quasilinear Schrödinger equation, ground state solutions, existence, concentration behavior

Abstract: In this article, we study the generalized quasilinear Schrödinger equation \begin{equation*} -\text{div}(\varepsilon^2g^2(u)\nabla u)+\varepsilon^2g(u)g'(u)|\nabla u|^2+V(x)u=K(x)|u|^{p-2}u,\,\, x\in\mathbb{R}^N, \end{equation*} where $N\geq3$, $\varepsilon>0$, $4 < p < 22^*$, $g\in\mathcal{C}^1(\mathbb{R},\mathbb{R}^{+})$, $V\in \mathcal{C}(\mathbb{R}^N)\cap L^\infty(\mathbb{R}^N)$ has a positive global minimum, and $K\in \mathcal{C}(\mathbb{R}^N)\cap L^\infty(\mathbb{R}^N)$ has a positive global maximum. By using a change of variable, we obtain the existence and concentration behavior of ground state solutions for this problem and establish a phenomenon of exponential decay.

Key words: generalized quasilinear Schrödinger equation, ground state solutions, existence, concentration behavior

中图分类号:

35J60

陈建华, 黄先玖, 程毕陶, 唐先华. EXISTENCE AND CONCENTRATION BEHAVIOR OF GROUND STATE SOLUTIONS FOR A CLASS OF GENERALIZED QUASILINEAR SCHRÖDINGER EQUATIONS IN $\mathbb{R}^N$[J]. 数学物理学报(英文版), 2020, 40(5): 1495-1524.

Jianhua CHEN, Xianjiu HUANG, Bitao CHENG, Xianhua TANG. EXISTENCE AND CONCENTRATION BEHAVIOR OF GROUND STATE SOLUTIONS FOR A CLASS OF GENERALIZED QUASILINEAR SCHRÖDINGER EQUATIONS IN $\mathbb{R}^N$[J]. Acta mathematica scientia,Series B, 2020, 40(5): 1495-1524.

参考文献 51

[1]	Aubin J P, Ekeland I. Applied Nonlinear Analysis, Pure and Apllied Mathematics. New York:John Wiley & Sons, Inc, 1984
[2]	Alves C O, Souto M A S. Existence of solutions for a class of nonlinear Schrödinger equations with potential vanishing at infinity. J Differential Equations 2013, 254:1977-1991
[5]	Bass F G, Nasanov N N. Nonlinear electromagnetic-spin waves. Phys Rep, 1990, 189:165-223
[4]	Bouard A D, Hayashi N, Saut J, Global existence of small solutions to a relativistic nonlinear Schrödinger equation. Comm Math Phys, 1997, 189:73-105
[5]	Brezis H, Nirenberge L. Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents. Commun Pure Appl Math, 1983, 36:437-477
[6]	Cassani D, do Ó J M, Moameni A. Existence and concentration of solitary waves for a class of quasilinear Schrödinger equations. Commun Pure Appl Anal, 2010, 9:281-306
[7]	Chen X L, Sudan R N. Necessary and sufficient conditions for self-focusing of short ultraintense laser pulse in underdense plasma. Phys Rev Lett, 1993, 70:2082-2085
[8]	Cheng Y, Yang J. Positive solution to a class of relativistic nonlinear Schrödinger equation. J Math Anal Appl, 2014, 411:665-674
[9]	Cheng Y, Yang J. Soliton solutions to a class of relativistic nonlinear Schrödinger equations. Appl Math Comput, 2015, 260:342-350
[10]	Chen J H, Tang X H, Cheng B T. Non-Nehari manifold method for a class of generalized quasilinear Schrödinger equations. Appl Math Lett, 2017, 74:20-26
[11]	Chen J H, Tang X H, Cheng B T. Ground states for a class of generalized quasilinear Schrödinger equations in RN. Mediterr J Math, 2017, 14:190
[12]	Chen J H, Tang X H, Cheng B T. Existence and nonexistence of positive solutions for a class of generalized quasilinear Schrodinger equations involving a Kirchhoff-type perturbation with critical Sobolev exponent. J Math Phys, 2018, 59:021505
[13]	Chen J H, Tang X H, Cheng B T. Ground state sign-changing solutions for a class of generalized quasilinear Schrodinger equations with a Kirchhoff-type perturbation. J Fixed Point Theory Appl, 2017, 19:3127-3149
[14]	Chen J H, Tang X H, Cheng B T. Existence and concentration of ground state solutions for a class of generalized quasilinear Schrödinger equation in R2. Preprint
[15]	Deng Y, Peng S, Yan S. Critical exponents and solitary wave solutions for generalized quasilinear Schrödinger equations. J Differ Equ, 2016, 260:1228-1262
[16]	Deng Y, Peng S, Yan S. Positive solition solutions for generalized quasilinear Schrödinger equations with critical growth. J Differ Equ, 2015, 258:115-147
[17]	Deng Y, Peng S, Wang J. Nodal soliton solutions for generalized quasilinear Schrödinger equations. J Math Phys, 2014, 55:051501
[18]	Deng Y, Peng S, Wang J. Nodal soliton solutions for quasilinear Schrödinger equations with critical exponent. J Math Phys, 2013, 54:011504
[19]	do Ó J M, Severo U. Solitary waves for a class of quasilinear Schrödinger equtions in dimension two. Calc Var Partial Differ Equ, 201038:375-315
[20]	Furtado M F, Silva E D, Silva M L. Existence of solutions for a generalized elliptic problem. J Math Phys, 2017, 58:031503
[21]	He X, Qian A, Zou W. Existence and concentration of positive solutions for quasilinear Schrödinger equations with critical growth. Nonlinearity, 2013, 26:3137-3168
[22]	Hasse R W. A general method for the solution of nonlinear soliton and kink Schrödinger equations. Z Phys B, 1980, 37:83-87
[23]	Kurihara S. Large-amplitude quasi-solitons in superfluid films. J Phys Soc Japan, 1981, 50:3262-3267
[24]	Li Q, Wu X. Existence of nontrivial solutions for generalized quasilinear Schrödinger equations with critical or supercritical growths. Acta Math Sci, 2017, 37B:1870-1880.
[25]	Lange H, Poppenberg M, Teismann H. Nash-Moser methods for the solution of quasilinear Schrödinger equations. Comm Partial Differ Equ, 1999, 24:1399-1418
[26]	Laedke E, Spatschek K, Stenflo L. Evolution theorem for a class of perturbed envelope soliton solutions. J Math Phys, 1983, 24:2764-2769
[27]	Liu J, Wang Z Q. Soliton solutions for quasilinear Schrödinger equations, I. Proc Amer Math Soc, 2003, 131:441-448
[28]	Liu J, Wang Y, Wang Z Q. Soliton solutions for quasilinear Schrödingerequations, Ⅱ. J Differ Equ, 2003, 187:47-493
[29]	Liu J, Wang Y, Wang Z Q. Solutions for quasilinear Schrödinger equations via the Nehari method. Comm Partial Differ Equ, 2004, 29:879-901
[30]	Cuccagna S. On instability of excited states of the nonlinear Schrödinger equation. Phys D, 2009, 238:38-54
[31]	Li F, Zhu X, Liang Z. Multiple solutions to a class of generalized quasilinear Schrödinger equations with a Kirchhoff-type perturbation. J Math Anal Appl, 2016, 443:11-38
[32]	Li Q, Teng K, Wu X. Ground state solutions and geometrically distinct solutions for generalized quasilinear Schrödinger equation. Math Methods Appl Sci, 2017, 40:2165-2176
[33]	Li Q, Wu X. Multiple solutions for generalized quasilinear Schrödinger equations. Math Methods Appl Sci, 2017, 40:1359-1366
[34]	Li Q, Wu X. Existence, multiplicity, and concentration of solutions for generalized quasilinear Schrödinger equations with critical growth. J Math Phys, 2017, 58:041501
[35]	Moameni A. Existence of solition solutions for a quasilinear Schrödinger equation involving critical exponent in RN. J Differ Equ, 2006, 229:570-587
[36]	Makhankov V G, Fedyanin V K. Nonlinear effects in quasi-one-dimensional models and condensed matter theory. Phys Rep, 1984, 104:1-86
[37]	Poppenberg M, Schmitt K, Wang Z Q. On the existence of soliton solutons to quasilinear Schrödinger equations. Calc Var Partial Differ Equ, 2002, 14:329-344
[38]	Pino M D, Felmer P L. Local mountain passes for semilinear elliptic problems in unbounded domains. Calc Var Partial Differ Equ, 1996, 4:121-137
[39]	Ritchie B. Relativistic self-focusing and channel formation in laser-plasma interaction. Phys Rev E, 1994, 50:687-689
[40]	Rabinowitz P H. On a class of nonlinear Schrödinger equations. Z Angew Math Phys, 1992, 43:270-291
[41]	Ruiz D, Siciliano G. Existence of ground states for a modified nonlinear Schrödinger equation. Nonlinearity, 2010, 23:1221-1233
[42]	Silva E A, Vieira G F. Quasilinear asymptotically periodic Schrödinger equations with critical growth. Calc Var Partial Differ Equ, 2010, 39:1-33
[43]	Shang X, Zhang J, Existence and concentration behavior of positive solutions for a quasilinear Schrödinger equation. J Math Anal Appl, 2014, 414:334-356
[44]	Shen Y, Wang Y. Standing waves for a class of quasilinear Schrödinger equations. Complex Var Elliptic Equ, 2016, 61:817-84
[45]	Shen Y, Wang Y. Two types of quasilinear elliptic equations with degenerate coerciveness and slightly superlinear growth. Appl Math Lett, 2015, 47:21-25
[46]	Shen Y, Wang Y. Soliton solutions for generalized quasilinear Schrödinger equations. Nonlinear Anal:Theory Methods Appl, 2013, 80:194-201
[47]	Wang W, Yang X, Zhao F K. Existence and concentration of ground state solutions for a subcubic quasilinear problem via Pohožaev manifold. J Math Anal Appl, 2015, 424:1471-1490
[48]	Wang Y, Zou W. Bound states to critical quasilinear Schrödinger equations. Nonlinear Differ Equ Appl, 2012, 19:19-47
[49]	Willem M. Minimax Theorems. Boston, MA:Birkhäuser, 1996
[50]	Yu Y, Zhao F K, Zhao L. The concentration behavior of ground state solutions for a fractional SchrödingerPoisson system. Calc Var Partial Differ Equ, 2017, 56:116
[51]	Zhu X, Li F, Liang Z. Existence of ground state solutions to a generalized quasilinear Schrödinger-Maxwell system. J Math Phys, 2016, 57:101505

[1]	酒全森, 王凤超. LOCAL EXISTENCE AND UNIQUENESS OF STRONG SOLUTIONS TO THE TWO DIMENSIONAL NONHOMOGENEOUS INCOMPRESSIBLE PRIMITIVE EQUATIONS[J]. 数学物理学报(英文版), 2020, 40(5): 1316-1334.
[2]	蒋伟峰, 王振. THE EXISTENCE OF A BOUNDED INVARIANT REGION FOR COMPRESSIBLE EULER EQUATIONS IN DIFFERENT GAS STATES[J]. 数学物理学报(英文版), 2020, 40(5): 1229-1239.
[3]	黄健骏, 姜正禄. GLOBAL EXISTENCE FOR THE RELATIVISTIC ENSKOG EQUATIONS[J]. 数学物理学报(英文版), 2020, 40(5): 1335-1351.
[4]	杜瑶, 唐春雷. GROUND STATE SOLUTIONS FOR A SCHRÖDINGER-POISSON SYSTEM WITH UNCONVENTIONAL POTENTIAL[J]. 数学物理学报(英文版), 2020, 40(4): 934-944.
[5]	黄文涛, 王莉. GROUND STATE SOLUTIONS OF NEHARI-POHOZAEV TYPE FOR A FRACTIONAL SCHRÖ DINGER-POISSON SYSTEM WITH CRITICAL GROWTH[J]. 数学物理学报(英文版), 2020, 40(4): 1064-1080.
[6]	李红民, 肖跃龙. LOCAL WELL-POSEDNESS OF STRONG SOLUTIONS FOR THE NONHOMOGENEOUS MHD EQUATIONS WITH A SLIP BOUNDARY CONDITIONS[J]. 数学物理学报(英文版), 2020, 40(2): 442-456.
[7]	Syed Sabyel HAIDER, Mujeeb Ur REHMAN. ULAM-HYERS-RASSIAS STABILITY AND EXISTENCE OF SOLUTIONS TO NONLINEAR FRACTIONAL DIFFERENCE EQUATIONS WITH MULTIPOINT SUMMATION BOUNDARY CONDITION[J]. 数学物理学报(英文版), 2020, 40(2): 589-602.
[8]	于佳利, 尚亚东, 狄华斐. GLOBAL NONEXISTENCE FOR A VISCOELASTIC WAVE EQUATION WITH ACOUSTIC BOUNDARY CONDITIONS[J]. 数学物理学报(英文版), 2020, 40(1): 155-169.
[9]	Lamia DJEBARA, Salem ABDELMALEK, Samir BENDOUKHA. GLOBAL EXISTENCE AND ASYMPTOTIC BEHAVIOR OF SOLUTIONS FOR SOME COUPLED SYSTEMS VIA A LYAPUNOV FUNCTIONAL[J]. 数学物理学报(英文版), 2019, 39(6): 1538-1550.
[10]	E. M. ELSAYED, S. HARIKRISHNAN, K. KANAGARAJAN. ON THE EXISTENCE AND STABILITY OF BOUNDARY VALUE PROBLEM FOR DIFFERENTIAL EQUATION WITH HILFER-KATUGAMPOLA FRACTIONAL DERIVATIVE[J]. 数学物理学报(英文版), 2019, 39(6): 1568-1578.
[11]	高真圣, 梁言, 谭忠. A GLOBAL EXISTENCE RESULT FOR KORTEWEG SYSTEM IN THE CRITICAL L^P FRAMEWORK[J]. 数学物理学报(英文版), 2019, 39(6): 1639-1660.
[12]	陈化, 徐辉阳. GLOBAL EXISTENCE, EXPONENTIAL DECAY AND BLOW-UP IN FINITE TIME FOR A CLASS OF FINITELY DEGENERATE SEMILINEAR PARABOLIC EQUATIONS[J]. 数学物理学报(英文版), 2019, 39(5): 1290-1308.
[13]	Ahmed ALSAEDI, Bashir AHMAD, Shorog ALJOUDI, Sotiris K. NTOUYAS. A STUDY OF A FULLY COUPLED TWO-PARAMETER SYSTEM OF SEQUENTIAL FRACTIONAL INTEGRO-DIFFERENTIAL EQUATIONS WITH NONLOCAL INTEGRO-MULTIPOINT BOUNDARY CONDITIONS[J]. 数学物理学报(英文版), 2019, 39(4): 927-944.
[14]	王玉柱, 田乃文. ON THE CAUCHY PROBLEM FOR IMBQ SYSTEM ARISING FROM DNA[J]. 数学物理学报(英文版), 2019, 39(4): 1136-1148.
[15]	Phuong LE, Vu HO. LIOUVILLE RESULTS FOR STABLE SOLUTIONS OF QUASILINEAR EQUATIONS WITH WEIGHTS[J]. 数学物理学报(英文版), 2019, 39(2): 357-368.

EXISTENCE AND CONCENTRATION BEHAVIOR OF GROUND STATE SOLUTIONS FOR A CLASS OF GENERALIZED QUASILINEAR SCHRÖDINGER EQUATIONS IN $\mathbb{R}^N$

EXISTENCE AND CONCENTRATION BEHAVIOR OF GROUND STATE SOLUTIONS FOR A CLASS OF GENERALIZED QUASILINEAR SCHRÖDINGER EQUATIONS IN $\mathbb{R}^N$

PDF (PC)

摘要/Abstract

引用本文

使用本文

参考文献 51

相关文章 15

Metrics

本文评价

推荐阅读 2