Topological Methods in Nonlinear Analysis

Infinitely many solutions for quasilinear Schrödinger equations under broken symmetry situation

Liang Zhang, Xianhua Tang, and Yi Chen

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


In this paper, we study the existence of infinitely many solutions for the quasilinear Schrödinger equations $$ \begin{cases} -\Delta u-\Delta(|u|^{\alpha})|u|^{\alpha-2}u=g(x,u)+h(x,u) &\text{for } x\in \Omega,\\ u=0 &\text{for } x\in \partial\Omega, \end{cases} $$ where $\alpha\geq 2$, $g, h\in C(\Omega\times \mathbb{R}, \mathbb{R})$. When $g$ is of superlinear growth at infinity in $u$ and $h$ is not odd in $u$, the existence of infinitely many solutions is proved in spite of the lack of the symmetry of this problem, by using the dual approach and Rabinowitz perturbation method. Our results generalize some known results and are new even in the symmetric situation.

Article information

Topol. Methods Nonlinear Anal., Volume 48, Number 2 (2016), 539-554.

First available in Project Euclid: 21 December 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Zhang, Liang; Tang, Xianhua; Chen, Yi. Infinitely many solutions for quasilinear Schrödinger equations under broken symmetry situation. Topol. Methods Nonlinear Anal. 48 (2016), no. 2, 539--554. doi:10.12775/TMNA.2016.057.

Export citation


  • S. Adachi and T. Watanabe, Uniqueness of the ground state solutions of quasilinear Schrödinger equations, Nonlinear Anal. 75 (2012), 819–833.
  • P. Bartolo, V. Benci and D. Fortunato, Abstract critical point theorems and application to some nonlinear problems with strong reasonce at infinity, Nonlinear Anal. 7 (1983), 981–1012.
  • R. Bartolo, A.M. Candela and A. Salvatore, Infinitely many radial solutions of a non-homogeneous problem, Discrete Contin. Dyn. Syst. Suppl. (2013), 51–59.
  • P. Bolle, On the Bolza problem, J. Differential Equations 152 (1999), 274–288.
  • P. Bolle, N. Ghoussoub and H. Tehrani, The multiplicity of solutions in nonhomogeneous boundary boundary value problems, Manuscripta Math. 101 (2002), 325–350.
  • A.M. Candela, G. Palmieri and A. Salvatore, Radial solutions of semilinear elliptic equations with broken symmetry, Topol. Methods Nonlinear Anal. 27 (2006), 117–132.
  • X.D. Fang and A. Szulkin, Multiple solutions for quasilinear Schrödinger equation, J. Differential Equations 254 (2013), 2015–2032.
  • S. Kurihara, Large-amplitude quasi-solitons in superfluid films, J. Phys. Soc. Japan 50 (1981), 3262–3267.
  • E.W. Laedke, K.H. Spatschek and L. Stenflo, Evolution theorem for a class of perturbed envelope soliton solutions, J. Math. Phys. 24 (1983), 2764–2769.
  • J.Q. Liu and Z.Q. Wang, Soliton solutions for quasilinear Schrödinger equations I, Proc. Amer. Math. Soc. 131 (2003), 441–448.
  • J.Q. Liu, Y. Wang and Z.Q. Wang, Solutions for quasilinear Schrödinger equations via the Nehari method, Comm. Partial Differential Equations 29 (2004), 879–892.
  • X.Q. Liu, J.Q. Liu and Z.Q. Wang, Quasilinear elliptic equations with critical growth via perturbation method, J. Differential Equations 254 (2013), 102–124.
  • ––––, Quasilinear elliptic equations via perturbation method, Proc. Amer. Math. Soc. 141 (2013), 253–263.
  • X.Q. Liu and F.K. Zhao, Existence of infinitely many solutions for quasilinear elliptic equations perturbed from symmetry, Adv. Nonlinear Studies 13 (2013), 965–978.
  • A. Nakamura, Damping and modification of exciton solitary waves, J. Phys. Soc. Japan 42 (1977), 1824–1835.
  • P. Rabinowitz, Multiple critical points of perturbed symmetric functionals, Trans. Amer. Math. Soc. 272 (1982), 753–769.
  • ––––, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Reg. Conf. Ser. in Math., vol. 65, Amer. Math. Soc., Providence, RI, 1986.
  • D. Ruiz and G. Siciliano, Existence of ground states for a modified nonlinear Schrödinger equation, Nonlinearity 23 (2010), 1221–1233.
  • A. Salvatore, Multiple solutions for perturbed elliptic equations in unbounded domains, Adv. Nonlinear Studies 3 (2003), 1–23.
  • M. Schechter and W. Zou, Infinitely many solutions to perturbed elliptic equations, J. Funct. Anal. 228 (2005), 1–38.
  • M. Struwe, Infinitely many critical points for functionals which are not even and applications to superlinear boundary value problems, Manuscripta Math. 32 (1980), 335–364.
  • H.T. Tehrani, Infinitely many solutions for indefinite semilinear elliptic equations without symmetry, Comm. Partial Differential Equations 21 (1996), 541–557.
  • X. Wu, Multiple solutions for quasilinear Schrödinger equations with a parameter, J. Differential Equations 256 (2014), 2619–2632.
  • X. Wu and K. Wu, Existence of positive solutions, negative solutions and high energy solutions for quasilinear elliptic equations on $\R^N$, Nonlinear Anal. RWA 16 (2014), 48–64.
  • J. Zhang, X.H. Tang and W. Zhang, Infinitely many solutions of quasilinear Schrödinger equation with sign-changing potential, J. Math. Anal. Appl. 420 (2014), 1762–1775.
  • ––––, Existence of infinitely many solutions for a quasilinear elliptic equation, Appl. Math. Lett. 37 (2014), 131–135.