Topological Methods in Nonlinear Analysis

On the Schrödinger equations with a nonlinearity in the critical growth

Jian Zhang

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

Abstract

In this paper, we consider the Schrödinger equation with a nonlinearity in the critical growth. The purpose of this paper is to establish the existence of ground states via variational methods.

Article information

Source
Topol. Methods Nonlinear Anal., Volume 44, Number 2 (2014), 457-469.

Dates
First available in Project Euclid: 11 April 2016

Permanent link to this document
https://projecteuclid.org/euclid.tmna/1460381365

Mathematical Reviews number (MathSciNet)
MR3328351

Citation

Zhang, Jian. On the Schrödinger equations with a nonlinearity in the critical growth. Topol. Methods Nonlinear Anal. 44 (2014), no. 2, 457--469. https://projecteuclid.org/euclid.tmna/1460381365


Export citation

References

  • C.O. Alves, M.A.S. Souto and M. Montenegro, Existence of a ground state solution for a nonlinear scalar field equation with critical growth, Calc. Var. Partial Differential Equations 43 (2012), 537–554.
  • A. Ambrosetti, V. Felli and A. Malchiodi, Ground states of nonlinear Schrödinger equations with potentials vanishing at infinity, J. Eur. Math. Soc. 7 (2005), 117–144.
  • A. Ambrosetti and P.H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973), 349–381.
  • T. Bartsch and Z.-Q. Wang, Existence and multiplicity results for some superlinear elliptic problems on $\mathbb{R}^N$, Comm. Partial Differential Equations 20 (1995), 1725–1741.
  • ––––, Multiple positive solutions for a nonlinear Schrödinger equations, Z. Angew. Math. Phys. 51 (2000), 366–384.
  • T. Bartsch, A. Pankov and Z.-Q. Wang, Nonlinear Schrödinger equations with steep potential well, Comm. Contemp. Math. 4 (2001), 549–569.
  • H. Berestycki, T. Gallouët and O. Kavian, Equations de champs scalaires euclidiens non lin$\acute{e}$aire dans le plan, C.R. Acad. Sci. Paris Sér. I Math. 297 (1983), 307–310.
  • H. Berestycki and P. L. Lions, Nonlinear scalar field equations \romI. Existence of a ground state, Arch. Ration. Mech. Anal. 82 (1983), 313–346.
  • H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math. 36 (1983), 437–477.
  • M. Clapp and Y.H. Ding, Positive solutions of a Schrödinger equation with critical nonlinearity, Z. Angew. Math. Phys. 55 (2004), 592–605.
  • J. Chabrowski and A. Szulkin, On a semilinear Schrödinger equation with critical Sobolev exponent, Proc. Amer. Math. Soc. 130 (2001), 85–93.
  • J. Chabrowski and J. Yang, Existence theorems for the Schrödinger equation involving a critical Sobolev exponent, Z. Angew. Math. Phys. 49 (1998), 276–293.
  • ––––, On Schrödinger equation with periodic potential and critical Sobolev exponent, Topol. Methods Nonlinear Anal. 12 (1998), 245–261.
  • V. Coti Zelati and P.H. Rabinowitz, Homoclinic type solutions for a semilinear elliptic \romPDE on $\mathbb{R}^N$, Comm. Pure Appl. Math. XIV (1992), 1217–1269.
  • Y.H. Ding and F.H. Lin, Solutions of perturbed Schrödinger equations with critical nonlinearity, Calc. Var. Partial Differential Equations 30 (2007), 231–249.
  • Y.H. Ding and K. Tanaka, Multiplicity of positive solutions of a nonlinear Schrödinger equation, Manuscripta Math. 112 (2003), 109–135.
  • L. Jeanjean, On the existence of bounded Palais–Smale sequences and application to a Landesman–Lazer-type problem set on $\mathbb{R}^N$, Proc. Roy. Soc. Edinburgh 129 (1999), 787–809.
  • L. Jeanjean and K. Tanaka, A positive solution for asymptotically linear elliptic problem on $\mathbb{R}^N$ autonomous at infinity, ESAIM Control Optim. Calc. Var. 7 (2002), 597–614.
  • ––––, A positive solution for a nonlinear Schrödinger equation on $\mathbb{R}^N$, Indiana Univ. Math. J., 54 (2005), 443–464.
  • Y.Q. Li, Z.-Q. Wang and J. Zeng, Ground states of nonlinear Schrödinger equations with potentials, Ann. Inst. H. Poincaré Anal. Non Linéaire 23 (2006), 829–837.
  • C. Liu, Z. Wang and H. Zhou, Asymptotically linear Schrödinger equations with potential vanishing at infinity, J. Differential Equations 245 (2008), 201–222.
  • P.H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys. 43 (1992), 270–291.
  • M. Schechter and W. Zou, Weak linking theorems and Schrödinger equations with critical Sobolev exponent, ESAIM: Control, Optimisation and Calculus of Variations 9 (2003), 601–619.
  • W.A. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys. 55 (1977), 149–162.
  • A. Szulkin and T. Weth, Ground state solutions for some indefinite variational problems, J. Funct. Anal. 257 (2009), 3802–3822.
  • F.A. van Heerden and Z.-Q. Wang, Schrödinger type equations with asymptotically linear nonlinearities, Differential Integral Equations, 16 (2003), 257–280.
  • F.A. van Heerden, Multiple solutions for a Schrödinger type equation with an asymptotically linear term, Nonlinear Anal. 55 (2003), 739–758.
  • M. Willem, Minimax Theorems, Birkhäuser, Boston (1996).
  • J. Zhang, On ground state solutions for quasilinear elliptic equations with a general nonlinearity in the critical growth, J. Math. Anal. Appl. 401 (2013), 232–241.
  • J. Zhang and W. Zou, The critical case for a Berestycki–Lions theorem, Sci. China Math. 57 (2014), 541–555.
  • J.J. Zhang and W. Zou, A Berestycki–Lions theorem revisited, Commun. Contemp. Math. 14 (2012), 125–133.