Existence of Weak Solutions for Generalized Quasilinear Schrödinger Equations

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• 作者：Hongxue Song ; Caisheng Chen
• 关键词：Quasilinear Schrödinger equation ; Mountain pass theorem ; Palais ; Smale sequence ; 35J20 ; 35J60
• 刊名：Journal of Dynamical and Control Systems
• 出版年：2016
• 出版时间：April 2016
• 年：2016
• 卷：22
• 期：2
• 页码：369-383
• 全文大小：270 KB
• 参考文献：1.Bass F, Nasanov NN. Nonlinear electromagnetic spin waves. Phys Reports 1990; 189:165–223.CrossRef
  2.Berestycki H, Gallouet T, Kavian O. Equations de Champs scalaires euclidiens non lineaires dans le plan. C R Acad Sci; Paris Ser I Math 1983;297(5):307–10.MathSciNet MATH
  3.Berestycki H, Gallouet T, Kavian O. 1984. Publications du Laboratoire d’Analyse Num6rique, Universit6 de Paris VI, MR 85e:35041.
  4.Berestycki H, Lions PL. Nonlinear scalar field equations I: existence of a state. Arch Rat Mech Anal 1983;82:313–46. MR 84h:35054a.MathSciNet MATH
  5.Chen XL, Sudan RN. Necessary and sufficient conditions for self-focusing of short ultraintense laser pulse. Phys Rev Lett 1993;70:2082–5.CrossRef
  6.Colin M. Stability of stationary waves for a quasilinear Schrödinger equation in space dimension 2. Adv in Diff Eqs 2003;8(1):1–28.MathSciNet MATH
  7.Colin M, Jeanjean L. Solutions for a quasilinear Schrödinger equation: a dual approach. Nonlinear Anal 2004;56:213–226.CrossRef MathSciNet MATH
  8.do Ó JM, de Medeiros ES. Remarks on least energy solutions for quasilinear elliptic problems in \(\mathbb {R}^{N}\) . Electron J Differential Equations 2003;83:14.
  9.do Ó JM, Miyagaki O, Soares S. Soliton solutions for quasilinear Schrödinger equations: the critical exponential case. Nonlinear Anal 2007;67:3357–3372.CrossRef MathSciNet MATH
  10.Evans LC. Partial differential equations, graduate studies in Mathematics. Amer Math Soc. 1998;19:261–76.
  11.Fleming WH. A selection-migration model in population genetic. J Math Bio 1975; 20:219–233.CrossRef MathSciNet
  12.Hasse RW. A general method for the solution of nonlinear soliton and kink Schrödinger equation. Z Phys B 1980;37:83–7.CrossRef MathSciNet
  13.Kelley PL. Self focusing of optical beams. Phys Rev Lett 1965;15:1005–8.CrossRef
  14.Kurihura S. Large-amplitude quasi-solitons in superfluids films. J Phys Soc Japan 1981;50:3262–7.CrossRef
  15.Laedke EW, Spatschek KH, Stenflo L. Evolution theorem for a class of perturbed envelope soliton solutions. J Math Phys 1983;24:2764–9.CrossRef MathSciNet MATH
  16.Lions PL. The concentration compactness principle in the calculus of variations, the locally compact case, parts I and II. Ann Inst H Poincaré Anal Non Lineairé 1984; 109–145:223–83.
  17.Liu J, Wang Y, Wang ZQ. Solutions for a quasilinear Schrödinger equation via the Nehari Method. Comm Partial Differential Equations 2004;29:879–901.CrossRef MathSciNet MATH
  18.Makhankov VG, Fedyanin VK. Non-linear effects in quasi-one-dimensional models of condensed matter theory. Phys Reports 1984;104:1–86.CrossRef MathSciNet
  19.Ritchie B. Relativistic self-focusing and channel formation in laser-plasma interactions. Phys Rev E 1994;50:687–9.CrossRef
  20.Schechter M. Linking Methods in Critical Point Theory. Boston: Birkhäuser; 1999.CrossRef MATH
  21.Serrin J. Local behavior of solutions of quasi-linear equations. Acta Math 1964; 111:247–302.CrossRef MathSciNet MATH
  22.Takeno S, Homma S. Classical planar Heinsenberg ferromagnet, complex scalar fields and nonlinear excitations. Progr Theoret Physics 1981;65:172–89.CrossRef MathSciNet
  23.Tolksdorff P. Regularity for a more general class of quasilinear elliptic equations. J Differential Equations 1984;51:126–50.CrossRef MathSciNet
  24.Trudinger NS. On Harnack type inequalities and their applications to quasilinear elliptic equations. Comm Pure Appl Math 1967;20:721–47.CrossRef MathSciNet MATH
  
• 作者单位：Hongxue Song (1) (2)
  Caisheng Chen (1)
  
  1. College of Science, Hohai University, Nanjing, 210098, People’s Republic of China
  2. College of Science, Nanjing University of Posts and Telecommunications, Nanjing, 210023, People’s Republic of China
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Calculus of Variations and Optimal Control
  Analysis
  Applications of Mathematics
  Systems Theory and Control
  
• 出版者：Springer Netherlands
• ISSN：1573-8698

文摘

This paper shows the existence of nontrivial weak solutions for the generalized quasilinear Schrödinger equations $$ -div(g^{p}(u)|\nabla u|^{p-2}\nabla u)+g^{p-1}(u)g^{\prime}(u)|\nabla u|^{p}+ V(x)|u|^{p-2}u=h(u),\,\, x\in \mathbb{R}^{N}, $$