一类无紧性扰动拟线性薛定谔方程的解
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摘要
利用Nehari流形方法研究了一类带有扰动项的拟线性薛定谔方程基态解的存在性。首先,利用一个代数方程证明了方程对应的Nehari流形是非空的。其次,根据流形的定义以及Sobolev不等式,证明了当限制在Nehari流形时元素范数有正下界。然后,利用集中紧性原理解决了工作空间紧性缺失的问题,进而得到方程对应泛函限制极小值的可达性。最后,利用条件极值原理得到方程基态解的存在性。
The existence of ground state solutions for a class of quasilinear Schr?dinger equations with disturbance term was studied through Nehari manifold method. Firstly,it was proved that the Nehari manifold corresponding to the equation was non-empty by using an algebraic equation. Secondly,according to the definition of manifold and the Sobolev inequality,the norm of the elements had a positive lower bound when the Nehari manifold was limited. And then,the lack problem of compactness in the working space was solved by the concentration compactness principle. Thus,the functional constraint minimum corresponding to the equation was obtained. Finally,the existence of the ground state solution of the equation was obtained by using the constrained extremum principle.
The existence of ground state solutions for a class of quasilinear Schr?dinger equations with disturbance term was studied through Nehari manifold method. Firstly,it was proved that the Nehari manifold corresponding to the equation was non-empty by using an algebraic equation. Secondly,according to the definition of manifold and the Sobolev inequality,the norm of the elements had a positive lower bound when the Nehari manifold was limited. And then,the lack problem of compactness in the working space was solved by the concentration compactness principle. Thus,the functional constraint minimum corresponding to the equation was obtained. Finally,the existence of the ground state solution of the equation was obtained by using the constrained extremum principle.
引文
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[17]DENG Y B,PENG S J,YAN S S.Critical exponents and solitary wave solutions for generalized quasilinear Schr9dinger equations[J].Journal of differential equations,2016,260(2):1228-1262.
[18]LIONS P L.The concentration-compactness principle in the calculus of variations.The locally compact case,part 1[J].Annales de l’institut henri poincare c,analys non linéaire,1984,1(2):109-145.
[19]LI G B,YE H Y.Existence of positive ground state solutions for the nonlinear Kirchhoff type equations in R3[J].Journal of differential equations,2014,257(2):566-600.
[20]郭大钧.非线性泛函分析[M].3版.北京:高等教育出版社,2015.
[2]KARIHARA S.Large-amplitude quasi-solitons on super-fluid films[J].Journal of the physical society of Japan,1981,50:3262-3267.
[3]LIU X Q,LIU J Q,WANG Z Q.Quasilinear elliptic equations via perturbation method[J].Proceedings of the American mathematical society,2012,141(1):253-263.
[4]CHEN J H,TANG X H,CHENG B T.Existence of ground state solutions for quasilinear Schr9dinger equations with superquadratic condition[J].Applied mathematical letter,2018,79:27-33.
[5]FANG X D.Positive solutions for quasilinear equation in RN[J].Communications on pure and applied analysis,2017,16:1603-1615.
[6]ZHANG W,LIU X Q.Infinitely many sign-changing solutions for a quasilinear elliptic equation in RN[J].Journal of mathematical analysis and applications,2015,420(2):722-740.
[7]LIU J Q,WANG Z Q.Multiple solutions for quasilinear elliptic equations with a finite potential well[J].Journal of differential equations,2014,257(8):2874-2899.
[8]LIU J Q,WANG Z Q.Soliton solutions for quasilinear Schr9dinger equationsⅠ[J].Proceedings of the American mathematical society,2002,131:473-493.
[9]LIU J Q,WANG Z Q.Soliton solutions for quasilinear Schr9dinger equationsⅡ[J].Journal of differential equations,2003,187:441-448.
[10]SILVA E A B,VIEIRA G F.Quasilinear asymptotically periodic Schr9dinger equations with critical growth[J].Calculus of variations and partial differential equations,2010,39(1/2):1-33.
[11]SILVA E A B,VIEIRA G F.Quasilinear asymptotically periodic Schr9dinger equations with subcritical growth[J].Nonlinear analysis,2009,72(6):2935-2949.
[12]COLIN M,JEANJEAN L.Solutions for a quasilinear Schr9dinger equations:a dual approach[J].Nonlinear analysis,2003,56(2):213-226.
[13]FENG X J,ZHANG Y.Existence of non-trivial solution for a class of modified Schr9dinger-poisson equations via perturbation method[J].Journal of mathematical analysis and applications,2016,442(2):673-684.
[14]DAVID G C.On a class of elliptic systems in RN[J].Electronic journal of differential equations,1994,7:1-14.
[15]BARTSCH T,WANG Z Q.Existence and multiplicity results for some superlinear elliptic problems on RN[J].Communications in partial differential equations,1995,20(9/10):1725-1741.
[16]ZHU X P,CAO D M.The concentration-compactness principle in nonlinear elliptic equations[J].Acta mathematical scientia,1989,9(3):307-328.
[17]DENG Y B,PENG S J,YAN S S.Critical exponents and solitary wave solutions for generalized quasilinear Schr9dinger equations[J].Journal of differential equations,2016,260(2):1228-1262.
[18]LIONS P L.The concentration-compactness principle in the calculus of variations.The locally compact case,part 1[J].Annales de l’institut henri poincare c,analys non linéaire,1984,1(2):109-145.
[19]LI G B,YE H Y.Existence of positive ground state solutions for the nonlinear Kirchhoff type equations in R3[J].Journal of differential equations,2014,257(2):566-600.
[20]郭大钧.非线性泛函分析[M].3版.北京:高等教育出版社,2015.