A positive ground state solution for a class of asymptotically periodic Schrödinger equations. (English) Zbl 1443.35047
Summary: In this paper, by using the reformative conditions, a class of asymptotically periodic Schrödinger equations are studied. Via the variational method, a positive ground state solution is obtained.
MSC:
| 35J91 | Semilinear elliptic equations with Laplacian, bi-Laplacian or poly-Laplacian |
| 35B10 | Periodic solutions to PDEs |
| 35J20 | Variational methods for second-order elliptic equations |
References:
| [1] | Berestycki, H.; Lions, P. L., Nonlinear scalar field equations. I existence of a ground state, Arch. Ration. Mech. Anal., 82, 313-345 (1983) · Zbl 0533.35029 |
| [2] | Lions, P. L., The concentration-compactness principle in the calculus of variations. The locally compact case, part 1, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1-4, 109-145 (1984) · Zbl 0541.49009 |
| [3] | Strauss, W. A., Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55, 149-162 (1977) · Zbl 0356.35028 |
| [4] | Berestycki, H.; Lions, P. L., Nonlinear scalar field equations. II existence of infinitely many solutions, Arch. Ration. Mech. Anal., 82, 347-375 (1983) · Zbl 0556.35046 |
| [5] | Coti Zelati, V.; Rabinowitz, P. H., Homoclinic type solutions for a semilinear elliptic PDE on \(R^n\), Comm. Pure Appl. Math., 45, 1217-1269 (1992) · Zbl 0785.35029 |
| [6] | Kryszewski, W.; Szulkin, A., Generalized linking theorem with an application to semilinear Schrödinger equation, Adv. Differential Equations, 3, 441-472 (1998) · Zbl 0947.35061 |
| [7] | Li, Y.; Wang, Z.-Q.; Zeng, J., Ground states of nonlinear Schrödinger equations with potentials, Ann. Inst. H. Poincaré Anal. Non Linéaire, 23, 829-837 (2006) · Zbl 1111.35079 |
| [8] | Szulkin, A.; Weth, T., Ground state solutions for some indefinite variational problems, J. Funct. Anal., 257, 3802-3822 (2009) · Zbl 1178.35352 |
| [9] | Szulkin, A.; Weth, T., The Method of Nehari Manifold. Handbook of Nonconvex Analysis and Applications, 597-632 (2010), Int. Press: Int. Press Somerville, MA · Zbl 1218.58010 |
| [10] | Cerami, G., Some nonlinear elliptic problems in unbounded domains, Milan J. Math., 74, 47-77 (2006) · Zbl 1121.35054 |
| [11] | Silva, E. A.B.; Vieira, G. F., Quasilinear asymptotically periodic Schrödinger equations with subcritical growth, Nonlinear Anal., 72, 2935-2949 (2010) · Zbl 1185.35052 |
| [12] | de Marchi, R., Schrödinger equations with asymptotically periodic terms, Proc. Roy. Soc. Edinburgh, 145A, 745-757 (2015) · Zbl 1331.35136 |
| [13] | Liu, S., On superlinear problems without the Ambrosetti and Rabinowitz condition, Nonlinear Anal., 73, 788-795 (2010) · Zbl 1192.35074 |
| [14] | Rabinowitz, P. H., On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43, 270-291 (1992) · Zbl 0763.35087 |
| [15] | Willem, M., Minimax Theorems (1996), Birkhäuser · Zbl 0856.49001 |
| [16] | Schechter, M., A variation of the mountain pass lemma and applications, J. Lond. Math. Soc., 2, 44, 491-502 (1991) · Zbl 0756.35032 |
| [17] | Lions, P. L., The concentration-compactness principle in the calculus of variations, The locally compact case, part 2, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1-4, 223-283 (1984) · Zbl 0704.49004 |
| [18] | Liu, J.; Liao, J.-F.; Tang, C.-L., Existence of ground state solution for a class of Kirchhoff type equation in \(R^N\), Proc. Roy. Soc. Edinburgh Sect., 146A (2016) · Zbl 1346.35203 |
| [19] | Liu, J.; Wang, Y.; Wang, Z.-Q., Solutions for quasilinear Schrödinger equations via the Nehari method, Comm. Partial Differential Equations, 29, 879-901 (2004) · Zbl 1140.35399 |
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