Infinitely many solutions for quasilinear Schrödinger systems with finite and sign-changing potentials

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• 作者：Yuxia Guo ; Jianjun Nie
• 关键词：Quasilinear systems ; Infinitely many solutions ; Sign ; changing potentials
• 刊名：Zeitschrift f¨¹r angewandte Mathematik und Physik
• 出版年：2016
• 出版时间：April 2016
• 年：2016
• 卷：67
• 期：2
• 全文大小：721 KB
• 参考文献：1.Ambrosetti A., Wang Z.: Positive solutions to a class of quasilinear elliptic equations on \({\mathbb{R}}\) . Discrete Contin. Dyn. Syst. 9, 55–68 (2003)MathSciNet CrossRef MATH
  2.Hartmann B., Zakrzewski W.: Electrons on hexagonal lattices and applications to nanotubes. Phys. Rev. B Condens. Matter 68, 184–302 (2003)
  3.Alves C., Figueiredo G., Severo U.: Multiplicity of positive solutions for a class of quasilinear problems. Adv. Differ. Equ. 14, 911–942 (2009)MathSciNet MATH
  4.Kenig C., Ponce G., Vega L.: The Cauchy problem for quasilinear Schrödinger equations. Invent. Math. 158, 343–388 (2004)MathSciNet CrossRef MATH
  5.Silva E., Vieira G.: Quasilinear asymptotically periodic Schrödinger equations with critical growth. Calc. Var. Partial Differ. Equ. 39, 1–33 (2010)MathSciNet CrossRef MATH
  6.Lin F., Silva E.: Quasilinear asymptotically periodic elliptic equations with critical growth. Nonlinear Anal. 71, 2890–2905 (2009)MathSciNet CrossRef MATH
  7.Cerami G., Devillanova G., Solimini S.: Infinitely many bound states solutions for some nonlinear scalar field equations. Calc. Var. Partial Differ. Equ. 23, 139–168 (2005)MathSciNet CrossRef MATH
  8.Lange H., Poppenperg M., Teismann H.: Nash-Moser methods for the solution of quasilinear Schrödinger equations. Commun. Partial Differ. Equ. 24, 1399–1418 (1999)CrossRef MATH
  9.Aires J., Souto M.: Existence of solutions for a quasilinear Schrödinger equation with vanishing potentials. J. Math. Anal. Appl. 416, 924–946 (2014)MathSciNet CrossRef MATH
  10.do Ó J., Miyagaki O., Soares S.: Soliton solutions for quasilinear Schrödinger equations with critical growth. J. Differ. Equ. 248, 722–744 (2010)CrossRef MATH
  11.do Ó J., Severo U.: Solitary waves for a class of quasilinear Schrödinger equations in dimension two. Calc. Var. Partial Differ. Equ. 38, 275–315 (2010)CrossRef MATH
  12.Nie J.J., Wu X., Zhu M.F.: Existence and multiplicity of non-trivial solutions for a class of modified Schrödinger equation with non-coercive potential. Appl. Math. Comput. 225, 677–694 (2013)MathSciNet CrossRef
  13.Liu J., Liu X., Wang Z.: Multiple Sign-Changing Solutions for Quasilinear Elliptic Equations via Perturbation Method. Commun. Partial Differ. Equ. 39, 2216–2239 (2014)MathSciNet CrossRef MATH
  14.Liu J., Wang Y., Wang Z.: Soliton solutions for quasilinear Schrödinger equations II. J. Differ. Equ. 187, 473–493 (2003)CrossRef MATH
  15.Liu J., Wang Y., Wang Z.: Solutions for quasilinear Schrödinger equations via the Nehari manifold. Commun. Partial Differ. Equ. 29, 879–901 (2004)CrossRef MATH
  16.Liu J., Wang Z.: Soliton solutions for quasilinear Schrödinger equation I. Proc. Am. Math. Soc. 131, 441–448 (2003)CrossRef MATH
  17.Liu J., Wang Z.: Multiple solutions for quasilinear elliptic equations with a finite potential well. J. Differ. Equ. 257, 2874–2899 (2014)MathSciNet CrossRef MATH
  18.Zhang J., Tang X.H., Zhang W.: Infinitely many solutions of quasilinear Schrödinger equation with signchanging potential. J. Math. Anal. Appl. 420, 1762–1775 (2014)MathSciNet CrossRef MATH
  19.Brizhik L., Eremko A., Piette B., Zakrzewski W.: Static solutions of a D-dimensional modified nonlinear Schrödinger equation. Nonlinearity 16, 1481–1497 (2003)MathSciNet CrossRef MATH
  20.Brüll L., Lange H.: Solitary waves for quasilinear Schrödinger equations. Expo. Math. 4, 279–288 (1986)MATH
  21.Colin M., Jeanjean L., Squassina M.: Stability and instability results for standing waves of quasilinear Schrödinger equations. Nonlinearity 23, 1353–1385 (2010)MathSciNet CrossRef MATH
  22.Colin M., Jeanjean L.: Solutions for a quasilinear Schrödinger equation: a dual approach. Nonlinear Anal. 56, 213–226 (2004)MathSciNet CrossRef MATH
  23.Colin M.: On the local well-posedness of quasilinear Schrödinger equations in arbitrary space dimension. Commun. Partial Differ. Equ. 27, 325–354 (2002)MathSciNet CrossRef MATH
  24.Poppenberg M., Schmidt K., Wang Z.: On the existence of soliton solutions to quasilinear Schrödinger equations. Calc. Var. Partial Differ. Equ. 14, 329–344 (2002)CrossRef MATH
  25.Poppenberg M.: On the local well posedness of quasilinear Schrödinger equations in arbitrary space dimension. J. Differ. Equ. 172, 83–115 (2001)MathSciNet CrossRef MATH
  26.Rabinowitz, P.H.: Minimax Methods in Critical Point Theory with Applications to Differential Equations. CBMS Reg. Conf. Ser. Math., vol.65, AMS, Providence, RI, (1986)
  27.Solimini S.: A note on compactness-type properties with respect to Lorentz norms of bounded subsets of a Sobolev space. Ann. Inst. Henry Poincaré 3, 319–337 (1995)MathSciNet MATH
  28.Solimini, S., Tintarev, K.: Concentration analysis in Banach spaces. Commun. Contemp. Math. (2015), DOI: 10.​1142/​s021919971550038​8
  29.Zou W.M., Schechter M.: Critical Point Theory and its Applications. Springer, New York (2006)MATH
  30.Liu X., Liu J., Wang Z.: Ground states for quasilinear Schrödinger equations with critical growth. Calc. Var. Partial Differ. Equ. 46, 641–669 (2013)CrossRef MATH
  31.Liu X., Liu J., Wang Z.: Quasilinear elliptic equations with critical growth via perturbation method. J. Differ. Equ. 254, 102–124 (2013)MathSciNet CrossRef MATH
  32.Guo Y., Li B.: Solutions for quasilinear Schrödinger systems with critical exponents. Z. Angew. Math. Phys. 66, 517–546 (2015)MathSciNet CrossRef MATH
  
• 作者单位：Yuxia Guo (1)
  Jianjun Nie (1)
  
  1. Department of Mathematics, Tsinghua University, Beijing, 100084, People’s Republic of China
  
• 刊物主题：Theoretical and Applied Mechanics; Mathematical Methods in Physics;
• 出版者：Springer Basel
• ISSN：1420-9039

文摘

We consider the following quasilinear Schrödinger system in \({\mathbb{R}^N}\) with \({N\geq3}\): $$\left\{\begin{array}{l}\sum_{i,j=1}^{N}D_j(a_{ij}(u)D_i u)-\frac{1}{2} \sum_{i,j=1}^{N}D_s a_{ij}(u) D_i u D_j u-A(x) u+F_u(u,v)=0 \\ \sum_{i,j=1}^{N}D_j(a_{ij}(v)D_iv)-\frac{1}{2} \sum_{i,j=1}^{N}D_s a_{ij}(v) D_i v D_j v-B(x)v+F_v(u,v)=0,\end{array} \right.$$ (P)where \({D_i=\frac{\partial}{\partial x_i},\ \ D_s a_{ij}(s)=\frac{d}{ds}a_{ij}(s)}\), \({F(u,v)}\) is the coupling term, \({A(x)}\) and \({B(x)}\) are finite and sign-changing potential functions. Using an approximation scheme and \({q}\)-Laplacian regularization, we prove the existence of infinitely many solutions for system \({(P)}\). Keywords Quasilinear systems Infinitely many solutions Sign-changing potentials Mathematics Subject Classification 35J62 58E05 Page %P Close Plain text Look Inside Reference tools Export citation EndNote (.ENW) JabRef (.BIB) Mendeley (.BIB) Papers (.RIS) Zotero (.RIS) BibTeX (.BIB) Add to Papers Other actions Register for Journal Updates About This Journal Reprints and Permissions Share Share this content on Facebook Share this content on Twitter Share this content on LinkedIn Related Content Supplementary Material (0) References (32) References1.Ambrosetti A., Wang Z.: Positive solutions to a class of quasilinear elliptic equations on \({\mathbb{R}}\). Discrete Contin. Dyn. Syst. 9, 55–68 (2003)MathSciNetCrossRefMATH2.Hartmann B., Zakrzewski W.: Electrons on hexagonal lattices and applications to nanotubes. Phys. Rev. B Condens. Matter 68, 184–302 (2003)3.Alves C., Figueiredo G., Severo U.: Multiplicity of positive solutions for a class of quasilinear problems. Adv. Differ. Equ. 14, 911–942 (2009)MathSciNetMATH4.Kenig C., Ponce G., Vega L.: The Cauchy problem for quasilinear Schrödinger equations. Invent. Math. 158, 343–388 (2004)MathSciNetCrossRefMATH5.Silva E., Vieira G.: Quasilinear asymptotically periodic Schrödinger equations with critical growth. Calc. Var. Partial Differ. Equ. 39, 1–33 (2010)MathSciNetCrossRefMATH6.Lin F., Silva E.: Quasilinear asymptotically periodic elliptic equations with critical growth. Nonlinear Anal. 71, 2890–2905 (2009)MathSciNetCrossRefMATH7.Cerami G., Devillanova G., Solimini S.: Infinitely many bound states solutions for some nonlinear scalar field equations. Calc. Var. Partial Differ. Equ. 23, 139–168 (2005)MathSciNetCrossRefMATH8.Lange H., Poppenperg M., Teismann H.: Nash-Moser methods for the solution of quasilinear Schrödinger equations. Commun. Partial Differ. Equ. 24, 1399–1418 (1999)CrossRefMATH9.Aires J., Souto M.: Existence of solutions for a quasilinear Schrödinger equation with vanishing potentials. J. Math. Anal. Appl. 416, 924–946 (2014)MathSciNetCrossRefMATH10.do Ó J., Miyagaki O., Soares S.: Soliton solutions for quasilinear Schrödinger equations with critical growth. J. Differ. Equ. 248, 722–744 (2010)CrossRefMATH11.do Ó J., Severo U.: Solitary waves for a class of quasilinear Schrödinger equations in dimension two. Calc. Var. Partial Differ. Equ. 38, 275–315 (2010)CrossRefMATH12.Nie J.J., Wu X., Zhu M.F.: Existence and multiplicity of non-trivial solutions for a class of modified Schrödinger equation with non-coercive potential. Appl. Math. Comput. 225, 677–694 (2013)MathSciNetCrossRef13.Liu J., Liu X., Wang Z.: Multiple Sign-Changing Solutions for Quasilinear Elliptic Equations via Perturbation Method. Commun. Partial Differ. Equ. 39, 2216–2239 (2014)MathSciNetCrossRefMATH14.Liu J., Wang Y., Wang Z.: Soliton solutions for quasilinear Schrödinger equations II. J. Differ. Equ. 187, 473–493 (2003)CrossRefMATH15.Liu J., Wang Y., Wang Z.: Solutions for quasilinear Schrödinger equations via the Nehari manifold. Commun. Partial Differ. Equ. 29, 879–901 (2004)CrossRefMATH16.Liu J., Wang Z.: Soliton solutions for quasilinear Schrödinger equation I. Proc. Am. Math. Soc. 131, 441–448 (2003)CrossRefMATH17.Liu J., Wang Z.: Multiple solutions for quasilinear elliptic equations with a finite potential well. J. Differ. Equ. 257, 2874–2899 (2014)MathSciNetCrossRefMATH18.Zhang J., Tang X.H., Zhang W.: Infinitely many solutions of quasilinear Schrödinger equation with signchanging potential. J. Math. Anal. Appl. 420, 1762–1775 (2014)MathSciNetCrossRefMATH19.Brizhik L., Eremko A., Piette B., Zakrzewski W.: Static solutions of a D-dimensional modified nonlinear Schrödinger equation. Nonlinearity 16, 1481–1497 (2003)MathSciNetCrossRefMATH20.Brüll L., Lange H.: Solitary waves for quasilinear Schrödinger equations. Expo. Math. 4, 279–288 (1986)MATH21.Colin M., Jeanjean L., Squassina M.: Stability and instability results for standing waves of quasilinear Schrödinger equations. Nonlinearity 23, 1353–1385 (2010)MathSciNetCrossRefMATH22.Colin M., Jeanjean L.: Solutions for a quasilinear Schrödinger equation: a dual approach. Nonlinear Anal. 56, 213–226 (2004)MathSciNetCrossRefMATH23.Colin M.: On the local well-posedness of quasilinear Schrödinger equations in arbitrary space dimension. Commun. Partial Differ. Equ. 27, 325–354 (2002)MathSciNetCrossRefMATH24.Poppenberg M., Schmidt K., Wang Z.: On the existence of soliton solutions to quasilinear Schrödinger equations. Calc. Var. Partial Differ. Equ. 14, 329–344 (2002)CrossRefMATH25.Poppenberg M.: On the local well posedness of quasilinear Schrödinger equations in arbitrary space dimension. J. Differ. Equ. 172, 83–115 (2001)MathSciNetCrossRefMATH26.Rabinowitz, P.H.: Minimax Methods in Critical Point Theory with Applications to Differential Equations. CBMS Reg. Conf. Ser. Math., vol.65, AMS, Providence, RI, (1986)27.Solimini S.: A note on compactness-type properties with respect to Lorentz norms of bounded subsets of a Sobolev space. Ann. Inst. Henry Poincaré 3, 319–337 (1995)MathSciNetMATH28.Solimini, S., Tintarev, K.: Concentration analysis in Banach spaces. Commun. Contemp. Math. (2015), DOI: 10.​1142/​s021919971550038​8 29.Zou W.M., Schechter M.: Critical Point Theory and its Applications. Springer, New York (2006)MATH30.Liu X., Liu J., Wang Z.: Ground states for quasilinear Schrödinger equations with critical growth. Calc. Var. Partial Differ. Equ. 46, 641–669 (2013)CrossRefMATH31.Liu X., Liu J., Wang Z.: Quasilinear elliptic equations with critical growth via perturbation method. J. Differ. Equ. 254, 102–124 (2013)MathSciNetCrossRefMATH32.Guo Y., Li B.: Solutions for quasilinear Schrödinger systems with critical exponents. Z. Angew. Math. Phys. 66, 517–546 (2015)MathSciNetCrossRefMATH About this Article Title Infinitely many solutions for quasilinear Schrödinger systems with finite and sign-changing potentials Journal Zeitschrift für angewandte Mathematik und Physik 67:30 Online DateApril 2016 DOI 10.1007/s00033-016-0621-7 Print ISSN 0044-2275 Online ISSN 1420-9039 Publisher Springer International Publishing Additional Links Register for Journal Updates Editorial Board About This Journal Manuscript Submission Topics Theoretical and Applied Mechanics Mathematical Methods in Physics Keywords 35J62 58E05 Quasilinear systems Infinitely many solutions Sign-changing potentials Industry Sectors Aerospace Engineering Oil, Gas & Geosciences Authors Yuxia Guo (1) Jianjun Nie (1) Author Affiliations 1. Department of Mathematics, Tsinghua University, Beijing, 100084, People’s Republic of China Continue reading... To view the rest of this content please follow the download PDF link above.