## Topological Methods in Nonlinear Analysis

### On the Schrödinger equations with a nonlinearity in the critical growth

Jian Zhang

#### 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

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

#### 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.