## Infinitely many solutions for asymptotically linear periodic Hamiltonian elliptic systems.(English)Zbl 1189.35091

Summary: This paper is concerned with the following periodic Hamiltonian elliptic system
$\begin{cases} -\Delta\varphi+ V(x)\varphi= G_\psi(x,\varphi,\psi) &\text{in }\mathbb R^N,\\ -\Delta\psi+ V(x)\psi= G_\varphi(x,\varphi,\psi) &\text{in }\mathbb R^N,\\ \varphi(x)\to 0\;\text{ and }\;\psi(x)\to 0 &\text{as }|x|\to\infty. \end{cases}$
Assuming the potential $$V$$ is periodic and 0 lies in a gap of $$\sigma(-\Delta+V)$$, $$G(x,\eta)$$, is periodic in $$x$$ and asymptotically quadratic in $$\eta=(\varphi,\psi)$$, existence and multiplicity of solutions are obtained via variational approach.

### MSC:

 35J57 Boundary value problems for second-order elliptic systems 35J50 Variational methods for elliptic systems 35J47 Second-order elliptic systems
Full Text:

### References:

 [1] N. Ackermann, On a periodic Schrödinger equation with nonlinear superlinear part. Math. Z.248 (2004) 423-443. Zbl1059.35037 · Zbl 1059.35037 [2] N. Ackermann, A superposition principle and multibump solutions of periodic Schrödinger equations. J. Func. Anal.234 (2006) 277-320. Zbl1126.35057 · Zbl 1126.35057 [3] C.O. Alves, P.C. Carrião and O.H. Miyagaki, On the existence of positive solutions of a perturbed Hamiltonian system in \Bbb R N . J. Math. Anal. Appl.276 (2002) 673-690. Zbl1056.35060 · Zbl 1056.35060 [4] A.I. Ávila and J. Yang, On the existence and shape of least energy solutions for some elliptic systems. J. Diff. Eq.191 (2003) 348-376. Zbl1109.35325 · Zbl 1109.35325 [5] A.I. Ávila and J. Yang, Multiple solutions of nonlinear elliptic systems. Nonlinear Differ. Equ. Appl.12 (2005) 459-479. Zbl1146.35346 · Zbl 1146.35346 [6] T. Bartsch and D.G. De Figueiredo, Infinitely many solutions of nonlinear elliptic systems, in Progress in Nonlinear Differential Equations and Their Applications35, Birkhäuser, Basel/Switzerland (1999) 51-67. Zbl0922.35049 · Zbl 0922.35049 [7] T. Bartsch and Y. Ding, Deformation theorems on non-metrizable vector spaces and applications to critical point theory. Math. Nach.279 (2006) 1-22. Zbl1117.58007 · Zbl 1117.58007 [8] V. Benci and P.H. Rabinowitz, Critical point theorems for indefinite functionals. Inven. Math.52 (1979) 241-273. Zbl0465.49006 · Zbl 0465.49006 [9] V. Coti-Zelati and P.H. Rabinowitz, Homoclinic orbits for second order Hamiltonian systems possessing superquadratic potentials. J. Amer. Math. Soc.4 (1991) 693-727. Zbl0744.34045 · Zbl 0744.34045 [10] V. Coti-Zelati and P.H. Rabinowitz, Homoclinic type solutions for a semilinear elliptic PDE on \Bbb R N . Comm. Pure Appl. Math.45 (1992) 1217-1269. Zbl0785.35029 · Zbl 0785.35029 [11] D.G. De Figueiredo and Y.H. Ding, Strongly indefinite functionals and multiple solutions of elliptic systems. Trans. Amer. Math. Soc.355 (2003) 2973-2989. Zbl1125.35338 · Zbl 1125.35338 [12] D.G. De Figueiredo and P.L. Felmer, On superquadratic elliptic systems. Trans. Amer. Math. Soc.343 (1994) 97-116. · Zbl 0799.35063 [13] D.G. De Figueiredo and J. Yang, Decay, symmetry and existence of solutions of semilinear elliptic systems. Nonlinear Anal.33 (1998) 211-234. Zbl0938.35054 · Zbl 0938.35054 [14] D.G. De Figueiredo, J. Marcos do Ó and B. Ruf, An Orlicz-space approach to superlinear elliptic systems. J. Func. Anal.224 (2005) 471-496. Zbl1210.35081 · Zbl 1210.35081 [15] Y. Ding and L. Jeanjean, Homoclinic orbits for a non periodic Hamiltonian system. J. Diff. Eq.237 (2007) 473-490. Zbl1117.37032 · Zbl 1117.37032 [16] Y. Ding and F.H. Lin, Semiclassical states of Hamiltonian systems of Schrödinger equations with subcritical and critical nonlinearies. J. Partial Diff. Eqs.19 (2006) 232-255. Zbl1104.35051 · Zbl 1104.35051 [17] J. Hulshof and R.C.A.M. Van de Vorst, Differential systems with strongly variational structure. J. Func. Anal.114 (1993) 32-58. Zbl0793.35038 · Zbl 0793.35038 [18] W. Kryszewski and A. Szulkin, An infinite dimensional Morse theory with applications. Trans. Amer. Math. Soc.349 (1997) 3181-3234. Zbl0892.58015 · Zbl 0892.58015 [19] W. Kryszewski and A. Szulkin, Generalized linking theorem with an application to semilinear Schrödinger equations. Adv. Differential Equations3 (1998) 441-472. Zbl0947.35061 · Zbl 0947.35061 [20] G. Li and A. Szulkin, An asymptotically periodic Schrödinger equation with indefinite linear part. Comm. Contemp. Math.4 (2002) 763-776. Zbl1056.35065 · Zbl 1056.35065 [21] G. Li and J. Yang, Asymptotically linear elliptic systems. Comm. Partial Diff. Eq.29 (2004) 925-954. Zbl1140.35406 · Zbl 1140.35406 [22] A. Pistoia and M. Ramos, Locating the peaks of the least energy solutions to an elliptic system with Neumann boundary conditions. J. Diff. Eq.201 (2004) 160-176. · Zbl 1246.35089 [23] M. Reed and B. Simon, Methods of Modern Mathematical Physics, IV Analysis of Operators. Academic Press, New York (1978). · Zbl 0401.47001 [24] E. Séré, Existence of infinitely many homoclinic orbits in Hamiltonian stysems. Math. Z.209 (1992) 133-160. Zbl0725.58017 · Zbl 0725.58017 [25] B. Sirakov, On the existence of solutions of Hamiltonian elliptic systems in RN. Adv. Differential Equations5 (2000) 1445-1464. Zbl1213.35223 · Zbl 1213.35223 [26] C. Troestler and M. Willem, Nontrivial solution of a semilinear Schrödinger equation. Comm. Partial Diff. Eq.21 (1996) 1431-1449. · Zbl 0864.35036 [27] M. Willem, Minimax Theorems. Birkhäuser, Berlin (1996). · Zbl 0856.49001 [28] J. Yang, Nontrivial solutions of semilinear elliptic systems in \Bbb R N . Electron. J. Diff. Eqns.6 (2001) 343-357. · Zbl 1099.35514
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.