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Abstract

In this paper, we consider the following Schrödinger equation $\begin{matrix} (0.1) & \{\begin{matrix} - Δ u - μ \frac{u}{| x |^{2}} + V (x) u = K (x) | u |^{2^{*} - 2} u + f (x, u), x \in R^{N}, \\ u \in H^{1} (R^{N}), \end{matrix} \end{matrix}$ where $N ⩾ 4$ , $0 ⩽ μ < \overline{μ}$ , $\overline{μ} = \frac{{(N - 2)}^{2}}{4}$ , V is periodic in x, K and f are asymptotically periodic in x, we take advantage of the generalized Nehari manifold approach developed by Szulkin and Weth to look for the ground state solution of (0.1).

Keywords

Schrödinger equation generalized Nehari manifold asymptotically periodic

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References

Ansari,

S.M.

Vaezpour and

Hesaaraki, Existence of positive solution for nonlocal singular fourth order Kirchhoff equation with Hardy potential, Positivity 21 (2017), 1545–1562. doi:10.1007/s11117-017-0484-y.

Bartsch and

Y.H.

Ding, On a nonlinear Schrödinger equation with periodic potential, Math. Ann 313 (1999), 15–37. doi:10.1007/s002080050248.

Bartsch,

Liu and

Weth, Sign changing solutions of superlinear Schrödinger equations, Comm. Partial Differential Equations 29 (2004), 25–42. doi:10.1081/PDE-120028842.

D.M.

Cao and

S.J.

Peng, A note on the sign-changing solution to elliptic problems with critical Sobolev and Hardy terms, J. Differential Equations. 193 (2003), 424–434. doi:10.1016/S0022-0396(03)00118-9.

D.M.

Cao and

S.S.

Yan, Infinitely many solutions for an elliptic problem involving critical Sobolev growth and Hardy potential, Calc. Var 38 (2010), 471–501. doi:10.1007/s00526-009-0295-5.

Catrina and

Z.Q.

Wang, On the Caffarelli–Kohn–Nirenberg inequalities: Sharp constants, existence (and nonexistence), and symmetry of extermal functions, Comm. Pure Appl. Math. 53 (2000), 1–30. doi:10.1002/(SICI)1097-0312(200001)53:1<1::AID-CPA1>3.0.CO;2-U.

Chabrowski and

Szulkin, On a semilinear Schrödinger equation with critical Sobolev exponent, Proc. Amer. Math. Soc 130 (2001), 85–93. doi:10.1090/S0002-9939-01-06143-3.

S.W.

Chen and

S.J.

Li, An elliptic problem with critical exponent and positive Hardy potetial, Hindawi Publishing Corporation 2 (2004), 91–98.

Z.J.

Chen and

W.M.

Zou, On an elliptic problem with critical exponent and Hardy potential, J. Differential Equations. 252 (2012), 969–987. doi:10.1016/j.jde.2011.09.042.

10.

Deng,

Jin and

Peng, Solutions of Schrödinger equations with invere square potential and critical nonlinearity, J. Differential Equations 252 (2012), 1376–1398. doi:10.1016/j.jde.2012.05.009.

11.

W.M.

Frank,

D.J.

Land and

R.M.

Spector, Singular potentials, Rev. Morden Phys 43 (1971), 36–98.

12.

Lin,

He and

Tang, Existence and asymptotic behavior of ground state solutions for asymptotically linear Schrödinger equation with inverse square potential, Communications on Pure & Applied Analysis 18 (2019), 1547–1565. doi:10.3934/cpaa.2019074.

13.

H.F.

Lins and

E.A.B.

Silva, Quasilinear asymptotically periodic elliptic equations with critical growth, Nonlinear Anal 71 (2009), 2890–2905. doi:10.1016/j.na.2009.01.171.

14.

Ruiz and

Willem, Elliptic problems with critical exponents and Hardy potentials, J. Differential Equations 190 (2003), 524–538. doi:10.1016/S0022-0396(02)00178-X.

15.

Szulkin and

Weth, Ground state solutions for some indefinite variational problems, J. Funct. Anal 257 (2009), 3802–3822. doi:10.1016/j.jfa.2009.09.013.

16.

Szulkin and

Weth, The method of Nehari manifold, in: Handbook of Nonconvex Analysis and Applications,

D.Y.

Gao and

Motreanu, eds, International Press, Boston, 2010, pp. 597–632.

17.

X.H.

Tang, Non-Nehari-manifold method for asymptotically linear Schrödinger equation, J. Aust. Math. Soc 98 (2015), 104–116. doi:10.1017/S144678871400041X.

18.

Wei and

Du, Exact singular behavior of positive solutions to nonlinear elliptic equations with a Hardy potential, J. Differential Equations 262 (2017), 3864–3886. doi:10.1016/j.jde.2016.12.004.

19.

Willem, Minimax Theorems, Birkhäuser, Boston, MA, 1996.

20.

Yang,

Zhang and

Liu, Sign-changing solutions for p-biharmonic equations with Hardy potential in

R^{N}

, Acta Mathematica Scientia 37B (2017), 593–606. doi:10.1016/S0252-9602(17)30025-5.

21.

Zhang,

J.X.

Xu and

F.B.

Zhang, Positive ground states for asymptotically periodic Schrödinger–Poisson systems, Math. Methods Appl 36 (2013), 427–439. doi:10.1002/mma.2604.

22.