New conditions on nonlinearity for a periodic Schrödinger equation having zero as spectrum. (English) Zbl 1312.35103
J. Math. Anal. Appl. 413, No. 1, 392-410 (2014); corrigendum ibid. 415, No. 1, 496 (2014).
Summary: We consider the semilinear Schrödinger equation
\[
\begin{cases} -\Delta u+V(x)u=f(x,u) & \text{for }x\in\mathbb R^N,\\u(x)\to 0 & \text{as }|x|\to \infty,\end{cases}
\]
where \(f\) is a superlinear and subcritical nonlinearity. We mainly study the case when both \(V\) and \(f\) are periodic in \(x\) and 0 is a boundary point of a spectral gap of \(-\Delta+V\). We extend a linking theorem of W. Kryszewski and A. Szulkin [Adv. Differ. Equ. 3, No. 3, 441–472 (1998; Zbl 0947.35061)] and establish a new variational setting which is more suitable to the above case. We obtain two theorems on the existence of ground state solutions with mild assumptions on \(f\).
Editorial remark: See also the acknowledgement of priority (“corrigendum”) in [ibid. 415, No. 1, 496 (2014; Zbl 1316.35126)].
Editorial remark: See also the acknowledgement of priority (“corrigendum”) in [ibid. 415, No. 1, 496 (2014; Zbl 1316.35126)].
MSC:
| 35J61 | Semilinear elliptic equations |
| 35J20 | Variational methods for second-order elliptic equations |
References:
| [1] | Alama, A.; Li, Y. Y., On “multibump” bound states for certain semilinear elliptic equations, Indiana Univ. Math. J., 41, 983-1026 (1992) · Zbl 0796.35043 |
| [2] | Ambrosetti, A.; Rabinowitz, P. H., Dual variational methods in critical point theory and applications, J. Funct. Anal., 14, 349-381 (1973) · Zbl 0273.49063 |
| [3] | Bartsch, T.; Ding, Y., On a nonlinear Schrödinger equation with periodic potential, Math. Ann., 313, 15-37 (1999) · Zbl 0927.35103 |
| [4] | Bartsch, T.; Ding, Y., Deformation theorems on non-metrizable vector spaces and applications to critical point theory, Math. Nachr., 279, 1267-1288 (2006) · Zbl 1117.58007 |
| [5] | Bartsch, T.; Wang, Z.-Q., Existence and multiplicity results for some superlinear elliptic problems on \(R^N\), Comm. Partial Differential Equations, 20, 1725-1741 (1995) · Zbl 0837.35043 |
| [6] | Buffoni, B.; Jeanjean, L.; Stuart, C. A., Existence of nontrivial solutions to a strongly indefinite semilinear equation, Proc. Amer. Math. Soc., 119, 179-186 (1993) · Zbl 0789.35052 |
| [7] | Coti Zelati, V.; Rabinowitz, P. H., Homoclinic type solutions for a semilinear elliptic PDE on \(R^N\), Comm. Pure Appl. Math., XIV, 1217-1269 (1992) · Zbl 0785.35029 |
| [8] | Ding, Y., Variational Methods for Strongly Indefinite Problems (2007), World Scientific: World Scientific Singapore · Zbl 1133.49001 |
| [9] | Ding, Y.; Lee, C., Multiple solutions of Schrödinger equations with indefinite linear part and super or asymptotically linear terms, J. Differential Equations, 222, 137-163 (2006) · Zbl 1090.35077 |
| [10] | Ding, Y. H.; Luan, S. X., Multiple solutions for a class of nonlinear Schrödinger equations, J. Differential Equations, 207, 423-457 (2004) · Zbl 1072.35166 |
| [11] | Ding, Y.; Szulkin, A., Bound states for semilinear Schrödinger equations with sign-changing potential, Calc. Var. Partial Differential Equations, 29, 3, 397-419 (2007) · Zbl 1119.35082 |
| [12] | Edmunds, D. E.; Evans, W. D., Spectral Theory and Differential Operators (1987), Clarendon Press: Clarendon Press Oxford · Zbl 0628.47017 |
| [13] | Evéquoz, G.; Stuart, C. A., Hadamard differentiability and bifurcation, Proc. Roy. Soc. Edinburgh Sect. A, 137, 1249-1285 (2007) · Zbl 1134.35014 |
| [14] | Jeanjean, L., On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer type problem set on \(R^N\), Proc. Roy. Soc. Edinburgh Sect. A, 129, 787-809 (1999) · Zbl 0935.35044 |
| [15] | Kryszewski, W.; Szulkin, A., Generalized linking theorem with an application to a semilinear Schrödinger equation, Adv. Difference Equ., 3, 441-472 (1998) · Zbl 0947.35061 |
| [16] | Li, G. B.; Szulkin, A., An asymptotically periodic Schrödinger equation with indefinite linear part, Commun. Contemp. Math., 4, 763-776 (2002) · Zbl 1056.35065 |
| [17] | Li, Y. Q.; 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 |
| [18] | 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, 223-283 (1984) · Zbl 0704.49004 |
| [19] | Liu, S., On superlinear Schrödinger equations with periodic potential, Calc. Var. Partial Differential Equations, 45, 1-9 (2012) · Zbl 1247.35149 |
| [20] | Liu, Z. L.; Wang, Z.-Q., On the Ambrosetti-Rabinowitz superlinear condition, Adv. Nonlinear Stud., 4, 561-572 (2004) |
| [21] | 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 |
| [22] | Pankov, A., Periodic nonlinear Schrödinger equation with application to photonic crystals, Milan J. Math., 73, 259-287 (2005) · Zbl 1225.35222 |
| [23] | Rabinowitz, P. H., On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43, 270-291 (1992) · Zbl 0763.35087 |
| [24] | Reed, M.; Simon, B., Methods of Modern Mathematical Physics, vol. IV. Analysis of Operators (1978), Academic Press: Academic Press New York · Zbl 0401.47001 |
| [25] | Schechter, M., Minimax Systems and Critical Point Theory (2009), Birkhäuser: Birkhäuser Boston · Zbl 1186.35043 |
| [26] | Stuart, C. A., Bifurcation for some non-Fréchet differentiable problems, Nonlinear Anal., 69, 1011-1024 (2008) · Zbl 1157.47045 |
| [27] | Szulkin, A.; Weth, T., Ground state solutions for some indefinite variational problems, J. Funct. Anal., 257, 12, 3802-3822 (2009) · Zbl 1178.35352 |
| [28] | Tang, X. H., Infinitely many solutions for semilinear Schrödinger equations with sign-changing potential and nonlinearity, J. Math. Anal. Appl., 401, 407-415 (2013) · Zbl 1364.35103 |
| [29] | Tang, X. H., New super-quadratic conditions on ground state solutions for superlinear Schrödinger equation, Adv. Nonlinear Stud. (2013), in press |
| [30] | Troestler, C.; Willem, M., Nontrivial solution of a semilinear Schrödinger equation, Comm. Partial Differential Equations, 21, 1431-1449 (1996) · Zbl 0864.35036 |
| [31] | Willem, M., Minimax Theorems (1996), Birkhäuser: Birkhäuser Boston · Zbl 0856.49001 |
| [32] | Willem, M.; Zou, W. M., On a Schrödinger equation with periodic potential and spectrum point zero, Indiana Univ. Math. J., 52, 109-132 (2003) · Zbl 1030.35068 |
| [33] | Yang, M. B.; Chen, W. X.; Ding, Y. H., Solutions for periodic Schrödinger equation with spectrum zero and general superlinear nonlinearities, J. Math. Anal. Appl., 364, 404-413 (2010) · Zbl 1189.35108 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
