A NONTRIVIAL SOLUTION OF A QUASILINEAR ELLIPTIC EQUATION VIA DUAL APPROACH

doi:10.1007/s10473-019-0220-8

数学物理学报(英文版) ›› 2019, Vol. 39 ›› Issue (2): 580-596.doi: 10.1007/s10473-019-0220-8

A NONTRIVIAL SOLUTION OF A QUASILINEAR ELLIPTIC EQUATION VIA DUAL APPROACH

杨先勇¹, 张薇², 赵富坤³

1. School of Preparatory Education, Yunnan Minzu University, Kunming 650500, China;School of Mathematics and Statistics, Central south University, Changsha 410205, China;
2. Department of Mathematics, Yunnan University, Kunming 650500, China;
3. Department of Mathematics, Yunnan Normal University, Kunming 650500, China

收稿日期:2017-10-07 修回日期:2018-04-07 出版日期:2019-04-25 发布日期:2019-05-06
通讯作者: Fukun ZHAO E-mail:fukunzhao@163.com
作者简介:Xianyong YANG,ynyangxianyong@163.com;Wei ZHANG,weizyn@163.com
基金资助:
This work was supported partially by National Natural Science Foundation of China (11771385, 11661083) and the Youth Foundation of Yunnan Minzu University (2017QNo3).

A NONTRIVIAL SOLUTION OF A QUASILINEAR ELLIPTIC EQUATION VIA DUAL APPROACH

Xianyong YANG¹, Wei ZHANG², Fukun ZHAO³

1. School of Preparatory Education, Yunnan Minzu University, Kunming 650500, China;School of Mathematics and Statistics, Central south University, Changsha 410205, China;
2. Department of Mathematics, Yunnan University, Kunming 650500, China;
3. Department of Mathematics, Yunnan Normal University, Kunming 650500, China

Received:2017-10-07 Revised:2018-04-07 Online:2019-04-25 Published:2019-05-06
Contact: Fukun ZHAO E-mail:fukunzhao@163.com
Supported by:
This work was supported partially by National Natural Science Foundation of China (11771385, 11661083) and the Youth Foundation of Yunnan Minzu University (2017QNo3).

摘要/Abstract

摘要： In this article, we are concerned with the existence of solutions of a quasilinear elliptic equation in R^N which includes the so-called modified nonlinear Schrödinger equation as a special case. Combining the dual approach and the nonsmooth critical point theory, we obtain the existence of a nontrivial solution.

关键词: nontrivial solution, quasilinear elliptic equation, nonsmooth critical point theory, dual approach

Abstract: In this article, we are concerned with the existence of solutions of a quasilinear elliptic equation in R^N which includes the so-called modified nonlinear Schrödinger equation as a special case. Combining the dual approach and the nonsmooth critical point theory, we obtain the existence of a nontrivial solution.

Key words: nontrivial solution, quasilinear elliptic equation, nonsmooth critical point theory, dual approach

杨先勇, 张薇, 赵富坤. A NONTRIVIAL SOLUTION OF A QUASILINEAR ELLIPTIC EQUATION VIA DUAL APPROACH[J]. 数学物理学报(英文版), 2019, 39(2): 580-596.

Xianyong YANG, Wei ZHANG, Fukun ZHAO. A NONTRIVIAL SOLUTION OF A QUASILINEAR ELLIPTIC EQUATION VIA DUAL APPROACH[J]. Acta mathematica scientia,Series B, 2019, 39(2): 580-596.

参考文献

[1] Brüll L, Lange H. Solitary waves for quasilinear Schrödinger equations. Exposition Math, 1986, 4(3):279-288
[2] Canino A, Degiovanni M. Nonsmooth critical point theory and quasilinear elliptic equations//Topological Methods in Differential Equations and Inclusions. Netherlands:Springer, 1995:1-50
[3] Chen J H, Tang X H, Cheng B T. Existence and nonexistence of positive solutions for a class of generalized quasilinear Schrödinger equations involving Kirchhoff-type perturbation with critical Sobolev exponent. J Math Phys, 2018, 59(2):021505, 24
[4] Chen S T, Tang X H. Ground state solutions for generalized quasilinear Schrödinger equations with variable potentials and Berestycki-Lions nonlinearities. J Math Phys, 2018, 59(8):081508, 18
[5] Colin M, Jeanjean L. Solutions for a quasilinear Schrödinger equation:a dual approach. Nonlinear Anal, 2004, 56(2):213-226
[6] Deng Y B, Guo Y X, Liu J Q. Existence of solutions for quasilinear elliptic equations with Hardy potential. J Math Phys, 2016, 57(3):031503, 15
[7] Deng Y B, Peng S J, Yan S S. Critical exponents and solitary wave solutions for generalized quasilinear Schrödinger equations. J Differential Equations, 2016, 260(2):1228-1262
[8] do Ó J M B, Miyagaki O H, Soares S H M. Soliton solutions for quasilinear Schrödinger equations with critical growth. J Differential Equations, 2010, 248(4):722-744
[9] Deng Z Y, Huang Y S. On positive G-symmetric solutions of a weighted quasilinear elliptic equation with critical Hardy-Sobolev exponent. Acta Math Sci, 2014, 34B(5):1619-1633
[10] Gasiński L, Papageorgiou N S. Nonsmooth Critical Point Theory and Nonlinear Boundary Value Problems. Boca Raton, FL:Chapman & Hall/CRC, 2005
[11] Jabri Y. The Mountain Pass Theorem. Cambridge:Cambridge University Press, 2003
[12] Li Q Q, Wu X. Existence of nontrivial solutions for generalized quasilinear Schrödinger equations with critical or supercritical growths. Acta Math Sci, 2017, 37B(6):1870-1880
[13] Li Z X, Shen Y T. Nonsmooth critical point theorems and its applications to quasilinear Schrödinger equations. Acta Math Sci, 2016, 36B(1):73-86
[14] Kurihara S. Exact soliton solution for superfluid film dynamics. J Phys Soc Japan, 1981, 50(11):3801-3805
[15] Liu J Q, Liu X Q, Wang Z Q. Multiple sign-changing solutions for quasilinear elliptic equations via perturbation method. Comm Partial Differential Equations, 2014, 39(12):2216-2239
[16] Liu J Q, Wang Y Q, Wang Z Q. Soliton solutions for quasilinear Schrödinger equations Ⅱ. J Differential Equations, 2003, 187(2):473-493
[17] Liu J Q, Wang Y Q, Wang Z Q. Solutions for quasilinear Schrödinger equations via the Nehari method. Comm Partial Differential Equations, 2004, 29(5/6):879-901
[18] Liu J Q, Wang Y Q, Wang Z Q. Soliton solutions for quasilinear Schrödinger equations I. Proc Amer Math Soc, 2003, 131(2):441-448
[19] Liu J Q, Wang Z Q, Guo Y X. Multibump solutions for quasilinear elliptic equations. J Funct Anal, 2012, 262(9):4040-4102
[20] Liu J Q, Wang Z Q, Wu X. Multibump solutions for quasilinear elliptic equations with critical growth. J Math Phys, 2013, 54(12):121501, 31
[21] Liu X Q, Liu J Q, Wang Z Q. Quasilinear elliptic equations via perturbation method. Proc Amer Math Soc, 2013. 141(1):253-263
[22] Lu W D. Variational Methods in Differential Equations. Scientific Publishing House in China, 2002
[23] Shen Y T, Wang Y J. Soliton solutions for generalized quasilinear Schrödinger equations. Nonlinear Anal., 2013, 80:194-201
[24] Shi H X, Chen H B. Existence and multiplicity of solutions for a class of generalized quasilinear Schrödinger equations. J Math Anal Appl, 2017, 452(1):578-594
[25] Wang Y J, Zou W M. Bound states to critical quasilinear Schrödinger equations. Nonlinear Differ Equ Appl, 2012, 19(1):19-47
[26] Willem M. Minimax theorems//Progress in Nonlinear Differential Equations and their Applications Vol 24, Boston, MA:Birkhäuser Boston, Inc, 1996
[27] Wu K. Positive solutions of quasilinear Schrödinger equations with critical growth. Appl Math Lett, 2015, 45:52-57
[28] Wu K, Wu X. Multiplicity of solutions for a quasilinear elliptic equation. Acta Math Sci, 2016, 36B(2):549-559
[29] Wu X, Wu K. Existence of positive solutions, negative solutions and high energy solutions for quasi-linear elliptic equations on R^N. Nonlinear Anal:Real World Appl, 2014, 16:48-64
[30] Wu X, Wu K. Geometrically distinct solutions for quasilinear elliptic equations. Nonlinearity, 2014, 27(5):987-1001
[31] Yang R R, Zhang W, Liu X Q. Sign-changing solutions for p-biharmonic equations with Hardy potential in R^N. Acta Math Sci, 2017, 37B(3):593-606
[32] Zhang J, Tang X H, Zhang W. Infinitely many solutions of quasilinear Schrödinger equation with signchanging potential. J Math Anal Appl, 2014, 420(2):1762-1775

A NONTRIVIAL SOLUTION OF A QUASILINEAR ELLIPTIC EQUATION VIA DUAL APPROACH

A NONTRIVIAL SOLUTION OF A QUASILINEAR ELLIPTIC EQUATION VIA DUAL APPROACH

PDF (PC)

摘要/Abstract

引用本文

使用本文

参考文献

相关文章 15

Metrics

本文评价

推荐阅读 0

[1]	向长林. GRADIENT ESTIMATES FOR SOLUTIONS TO QUASILINEAR ELLIPTIC EQUATIONS WITH CRITICAL SOBOLEV GROWTH AND HARDY POTENTIAL[J]. 数学物理学报(英文版), 2017, 37(1): 58-68.
[2]	魏利, 陈蕊, Ravi P. AGARWAL, Patricia YJ WONG. EXISTENCE AND UNIQUENESS OF NON-TRIVIAL SOLUTION OF PARABOLIC p-LAPLACIAN-LIKE DIFFERENTIAL EQUATION WITH MIXED BOUNDARIES[J]. 数学物理学报(英文版), 2016, 36(6): 1780-1792.
[3]	李周欣, 沈尧天. NONSMOOTH CRITICAL POINT THEOREMS AND ITS APPLICATIONS TO QUASILINEAR SCHRÖDINGER EQUATIONS[J]. 数学物理学报(英文版), 2016, 36(1): 73-86.
[4]	王影, 王明新. SOLUTIONS TO NONLINEAR ELLIPTIC EQUATIONS WITH A GRADIENT[J]. 数学物理学报(英文版), 2015, 35(5): 1023-1036.
[5]	邓志颖, 黄毅生. ON POSITIVE G-SYMMETRIC SOLUTIONS OF A WEIGHTED QUASILINEAR ELLIPTIC EQUATION WITH CRITICAL HARDY-SOBOLEV EXPONENT[J]. 数学物理学报(英文版), 2014, 34(5): 1619-1633.
[6]	尚月赟, 王莉. MULTIPLE NONTRIVIAL SOLUTIONS FOR A CLASS OF SEMILINEAR POLYHARMONIC EQUATIONS[J]. 数学物理学报(英文版), 2014, 34(5): 1495-1509.
[7]	丁凌, 唐春雷. POSITIVE SOLUTIONS FOR CRITICAL QUASILINEAR ELLIPTIC EQUATIONS WITH MIXED DIRICHLET-NEUMANN BOUNDARY CONDITIONS[J]. 数学物理学报(英文版), 2013, 33(2): 443-470.
[8]	李工宝, 王春花. THE EXISTENCE OF NONTRIVIAL SOLUTIONS TO A SEMILINEAR ELLIPTIC SYSTEM ON R^N WITHOUT THE AMBROSETTI-RABINOWITZ CONDITION[J]. 数学物理学报(英文版), 2010, 30(6): 1917-1936.
[9]	Zdzislaw Denkowski, Leszek Gasinski, Nikolaos S. Papageorgiou. MULTIPLE SOLUTIONS FOR NONAUTONOMOUS SECOND ORDER PERIODIC SYSTEMS[J]. 数学物理学报(英文版), 2010, 30(1): 341-349.
[10]	李翀，刘春根. THE EXISTENCE OF NONTRIVIAL SOLUTIONS OF HAMILTONIAN SYSTEMS WITH LAGRANGIAN BOUNDARY CONDITIONS[J]. 数学物理学报(英文版), 2009, 29(2): 313-326.
[11]	姚仰新; 王荣鑫; 沈尧天. NONTRIVIAL SOLUTION FOR A CLASS OF SEMILINEAR BIHARMONIC EQUATION INVOLVING CRITICAL EXPONENTS[J]. 数学物理学报(英文版), 2007, 27(3): 509-514.
[12]	Michael Filippakis; Leszek Gasinski; Nikolaos S. Papageorgiou. SEMILINEAR HEMIVARIATIONAL INEQUALITIES WITH STRONG RESONANCE AT INFINITY[J]. 数学物理学报(英文版), 2006, 26(1): 59-73.
[13]	娄本东. SOLUTIONS FOR SUPERLINEAR (n &minus\|1, 1) CONJUGATE BOUNDARY VALUE PROBLEMS[J]. 数学物理学报(英文版), 2001, 21(2): 259-264.
[14]	谭忠, 姚正安. THE EXISTENCE OF MULTIPLE SOLUTIONS OF p-LAPLACIAN ELLIPTIC EQUATION[J]. 数学物理学报(英文版), 2001, 21(2): 203-212.
[15]	林长好. A PHRAGMEN-LINDELÖF ALTERNATIVE FOR A CLASS OF SECOND ORDER QUASILINEAR EQUATIONS IN R³[J]. 数学物理学报(英文版), 1996, 16(2): 181-191.