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Lin, Xiaoyan; Tang, X. H.

Nehari-type ground state positive solutions for superlinear asymptotically periodic Schrödinger equations. (English) Zbl 1474.35333

Abstr. Appl. Anal. 2014, Article ID 607078, 7 p. (2014).
Summary: We deal with the existence of Nehari-type ground state positive solutions for the nonlinear Schrödinger equation \(- \Delta u + V \left(x\right) u = f \left(x, u\right), x \in \mathbb{R}^N, u \in H^1 \left(\mathbb{R}^N\right)\). Under a weaker Nehari condition, we establish some existence criteria to guarantee that the above problem has Nehari-type ground state solutions by using a more direct method in two cases: the periodic case and the asymptotically periodic case.

Cited in 6 Documents

MSC:

35J91 Semilinear elliptic equations with Laplacian, bi-Laplacian or poly-Laplacian
35J20 Variational methods for second-order elliptic equations
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References:

[1] Alama, S.; Li, Y. Y., On “multibump” bound states for certain semilinear elliptic equations, Indiana University Mathematics Journal, 41, 4, 983-1026 (1992) · Zbl 0796.35043 · doi:10.1512/iumj.1992.41.41052
[2] Angenent, S., The shadowing lemma for elliptic PDE, Dynamics of Infinite Dimensional Systems. Dynamics of Infinite Dimensional Systems, NATO ASI Series, 37, 7-22 (1986), Berlin, Germany: Springer, Berlin, Germany · Zbl 0653.35030 · doi:10.1007/978-3-642-86458-2_2
[3] Bartsch, T.; Ding, Y., Homoclinic solutions of an infinite-dimensional Hamiltonian system, Mathematische Zeitschrift, 240, 2, 289-310 (2002) · Zbl 1008.37040 · doi:10.1007/s002090100383
[4] Bartsch, T.; Pankov, A.; Wang, Z., Nonlinear Schrödinger equations with steep potential well, Communications in Contemporary Mathematics, 3, 4, 549-569 (2001) · Zbl 1076.35037 · doi:10.1142/S0219199701000494
[5] Chabrowski, J.; Szulkin, A., On a semilinear Schrödinger equation with critical Sobolev exponent, Proceedings of the American Mathematical Society, 130, 1, 85-93 (2002) · Zbl 0984.35150 · doi:10.1090/S0002-9939-01-06143-3
[6] Deng, Y.; Jin, L.; Peng, S., Solutions of Schrödinger equations with inverse square potential and critical nonlinearity, Journal of Differential Equations, 253, 5, 1376-1398 (2012) · Zbl 1248.35058 · doi:10.1016/j.jde.2012.05.009
[7] Ding, Y.; Szulkin, A., Bound states for semilinear Schrödinger equations with sign-changing potential, Calculus of Variations and Partial Differential Equations, 29, 3, 397-419 (2007) · Zbl 1119.35082 · doi:10.1007/s00526-006-0071-8
[8] Ding, Y.; Luan, S. X., Multiple solutions for a class of nonlinear Schrödinger equations, Journal of Differential Equations, 207, 2, 423-457 (2004) · Zbl 1072.35166 · doi:10.1016/j.jde.2004.07.030
[9] Ding, Y.; Lee, C., Multiple solutions of Schrödinger equations with indefinite linear part and super or asymptotically linear terms, Journal of Differential Equations, 222, 1, 137-163 (2006) · Zbl 1090.35077 · doi:10.1016/j.jde.2005.03.011
[10] Kryszewski, W.; Szulkin, A., Generalized linking theorem with an application to a semilinear Schrödinger equation, Advances in Differential Equations, 3, 3, 441-472 (1998) · Zbl 0947.35061
[11] Kryszewski, W.; Szulkin, A., Infinite-dimensional homology and multibump solutions, Journal of Fixed Point Theory and Applications, 5, 1, 1-35 (2009) · Zbl 1189.58004 · doi:10.1007/s11784-009-0104-y
[12] Li, Y. Q.; Wang, Z. Q.; Zeng, J., Ground states of nonlinear Schrödinger equations with potentials, Annales de l’Institut Henri Poincaré (C) Analyse Non Linéaire, 23, 6, 829-837 (2006) · Zbl 1111.35079
[13] Liu, J. Q.; Wang, Y. Q.; Wang, Z. Q., Solutions for quasilinear Schrödinger equations via the Nehari method, Communications in Partial Differential Equations, 29, 5-6, 879-901 (2004) · Zbl 1140.35399 · doi:10.1081/PDE-120037335
[14] Liu, Z.; Wang, Z.-Q., On the Ambrosetti-Rabinowitz superlinear condition, Advanced Nonlinear Studies, 4, 561-572 (2004)
[15] Pankov, A., Periodic nonlinear Schrödinger equation with application to photonic crystals, Milan Journal of Mathematics, 73, 259-287 (2005) · Zbl 1225.35222 · doi:10.1007/s00032-005-0047-8
[16] Rabinowitz, P. H., Minimax Methods in Critical Point Theory with Applications to Differential Equations. Minimax Methods in Critical Point Theory with Applications to Differential Equations, Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics, 65 (1986), Providence, RI, USA: American Mathematical Society, Providence, RI, USA · Zbl 0609.58002
[17] Rabinowitz, P. H., On a class of nonlinear Schrödinger equations, Zeitschrift für Angewandte Mathematik und Physik, 43, 2, 270-291 (1992) · Zbl 0763.35087 · doi:10.1007/BF00946631
[18] Schechter, M., Minimax Systems and Critical Point Theory (2009), Boston, Mass, USA: Birkhäuser, Boston, Mass, USA · Zbl 1186.35043
[19] Schechter, M.; Zou, W., Double linking theorem and multiple solutions, Journal of Functional Analysis, 205, 1, 37-61 (2003) · Zbl 1232.35072 · doi:10.1016/S0022-1236(03)00265-9
[20] Szulkin, A.; Weth, T., Ground state solutions for some indefinite variational problems, Journal of Functional Analysis, 257, 12, 3802-3822 (2009) · Zbl 1178.35352 · doi:10.1016/j.jfa.2009.09.013
[21] Tang, X. H., Infinitely many solutions for semilinear Schrödinger equations with sign-changing potential and nonlinearity, Journal of Mathematical Analysis and Applications, 401, 1, 407-415 (2013) · Zbl 1364.35103 · doi:10.1016/j.jmaa.2012.12.035
[22] Tang, X. H., New super-quadratic conditions on ground state solutions for superlinear Schrödinger equation, Advanced Nonlinear Studies, 14, 349-361 (2014) · Zbl 1305.35036
[23] Tang, X. H., New conditions on nonlinearity for a periodic Schrödinger equation having zero as spectrum, Journal of Mathematical Analysis and Applications, 413, 1, 392-410 (2014) · Zbl 1312.35103 · doi:10.1016/j.jmaa.2013.11.062
[24] Troestler, C.; Willem, M., Nontrivial solution of a semilinear Schrödinger equation, Communications in Partial Differential Equations, 21, 9-10, 1431-1449 (1996) · Zbl 0864.35036 · doi:10.1080/03605309608821233
[25] van Heerden, F. A., Homoclinic solutions for a semilinear elliptic equation with an asymptotically linear nonlinearity, Calculus of Variations and Partial Differential Equations, 20, 4, 431-455 (2004) · Zbl 1142.35422 · doi:10.1007/s00526-003-0242-9
[26] Willem, M., Minimax Theorems (1996), Boston, Mass, USA: Birkhäuser, Boston, Mass, USA · Zbl 0856.49001 · doi:10.1007/978-1-4612-4146-1
[27] Yang, M., Ground state solutions for a periodic Schrödinger equation with superlinear nonlinearities, Nonlinear Analysis: Theory, Methods & Applications, 72, 5, 2620-2627 (2010) · Zbl 1184.35139 · doi:10.1016/j.na.2009.11.009
[28] Yang, M.; Chen, W.; Ding, Y., Solutions for periodic Schrödinger equation with spectrum zero and general superlinear nonlinearities, Journal of Mathematical Analysis and Applications, 364, 2, 404-413 (2010) · Zbl 1189.35108 · doi:10.1016/j.jmaa.2009.10.022
[29] Lions, P. L., The concentration-compactness principle in the calculus of variations. The locally compact case. I, Annales de l’Institut Henri Poincaré. Analyse Non Linéaire, 1, 109-145 (1984) · Zbl 0541.49009
[30] Struwe, M., Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems (2000), Berlin, Germany: Springer, Berlin, Germany · Zbl 0939.49001 · doi:10.1007/978-3-662-04194-9
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