## Solutions for a class of nonperiodic superquadratic Hamiltonian elliptic systems involving gradient terms.(English)Zbl 1375.35154

Summary: In the present paper, we consider the following Hamiltonian elliptic system (HES): $$- \Delta u + b \left(x\right) \cdot \nabla u + V \left(x\right) u = H_v \left(x, u, v\right)$$, $$x \in \mathbb{R}^N$$, $$- \Delta v - b \left(x\right) \cdot \nabla v + V \left(x\right) v = H_u \left(x, u, v\right)$$, $$x \in \mathbb{R}^N$$. A new existence result of nontrivial solutions is obtained for the system (HES) via variational methods for strongly indefinite problems, which generalizes some known results in the literatures.

### MSC:

 35J47 Second-order elliptic systems 35J50 Variational methods for elliptic systems

### Keywords:

Hamiltonian elliptic system; variational methods
Full Text:

### References:

 [1] Lions, J.-L., Optimal Control of Systems Governed by Partial Differential Equations, (1971), Berlin, Germany: Springer, Berlin, Germany · Zbl 0203.09001 [2] Itô, S., Diffusion Equations. Diffusion Equations, Translations of Mathematical Monographs, 114, (1992), Providence, RI, USA: American Mathematical Society, Providence, RI, USA [3] Nagasawa, M., Schrödinger Equations and Diffusion Theory. Schrödinger Equations and Diffusion Theory, Monographs in Mathematics, 86, (1993), Basel, Switzerland: Birkhäuser, Basel, Switzerland · Zbl 0780.60003 [4] De Figueiredo, D. G.; Yang, J., Decay, symmetry and existence of solutions of semilinear elliptic systems, Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal, 33, 3, 211-234, (1998) · Zbl 0938.35054 [5] Sirakov, B., On the existence of solutions of Hamiltonian elliptic systems in $$R^N$$, Advances in Differential Equations, 5, 10–12, 1445-1464, (2000) · Zbl 1213.35223 [6] Zhao, F. K.; Zhao, L. G.; Ding, Y. H., Multiple solutions for asymptotically linear elliptic systems, Nonlinear Differential Equations and Applications, 15, 6, 673-688, (2008) · Zbl 1170.35384 [7] Alves, C. O.; Carrião, P. C.; Miyagaki, O. H., On the existence of positive solutions of a perturbed Hamiltonian system in R, Journal of Mathematical Analysis and Applications, 276, 2, 673-690, (2002) · Zbl 1056.35060 [8] Ávila, A. I.; Yang, J., Multiple solutions of nonlinear elliptic systems, Nonlinear Differential Equations and Applications, 12, 4, 459-479, (2005) · Zbl 1146.35346 [9] Ávila, A. I.; Yang, J., On the existence and shape of least energy solutions for some elliptic systems, Journal of Differential Equations, 191, 2, 348-376, (2003) · Zbl 1109.35325 [10] Bartsch, T.; Ding, Y., Homoclinic solutions of an infinite-dimensional Hamiltonian system, Mathematische Zeitschrift, 240, 2, 289-310, (2002) · Zbl 1008.37040 [11] Bartsch, T.; De Figueiredo, D. G., Infinitely many solutions of nonlinear elliptic systems, Nonlinear Differential Equations and Applications, 35, 51-67, (1999) · Zbl 0922.35049 [12] Busca, J.; Sirakov, B., Symmetry results for semilinear elliptic systems in the whole space, Journal of Differential Equations, 163, 1, 41-56, (2000) · Zbl 0952.35033 [13] Lair, A. V.; Wood, A. W., Existence of entire large positive solutions of semilinear elliptic systems, Journal of Differential Equations, 164, 2, 380-394, (2000) · Zbl 0962.35052 [14] Li, G.; Yang, J., Asymptotically linear elliptic systems, Communications in Partial Differential Equations, 29, 5-6, 925-954, (2004) · Zbl 1140.35406 [15] Pistoia, A.; Ramos, M., Locating the peaks of the least energy solutions to an elliptic system with Neumann boundary conditions, Journal of Differential Equations, 201, 1, 160-176, (2004) · Zbl 1246.35089 [16] Schechter, M.; Zou, W. M., Homoclinic orbits for Schrödinger systems, Michigan Mathematical Journal, 51, 1, 59-71, (2003) · Zbl 1195.35281 [17] Yang, J., Nontrivial solutions of semilinear elliptic systems in $$R^N$$, Electronic Journal of Differential Equations, 6, 343-357, (2001) · Zbl 1099.35514 [18] Wang, J.; Xu, J.; Zhang, F., Existence of solutions for nonperiodic superquadratic Hamiltonian elliptic systems, Nonlinear Analysis. Theory, Methods & Applications, 72, 3-4, 1949-1960, (2010) · Zbl 1183.35115 [19] Zhao, F.; Ding, Y., On Hamiltonian elliptic systems with periodic or non-periodic potentials, Journal of Differential Equations, 249, 12, 2964-2985, (2010) · Zbl 1205.35080 [20] Ding, Y. H., Variational Methods for Strongly Indefinite Problems, (2008), Singapore: World Scientific Press, Singapore [21] Ding, Y.; Lee, C., Existence and exponential decay of homoclinics in a nonperiodic superquadratic Hamiltonian system, Journal of Differential Equations, 246, 7, 2829-2848, (2009) · Zbl 1162.70014 [22] Ding, Y.; Wei, J., Stationary states of nonlinear Dirac equations with general potentials, Reviews in Mathematical Physics, 20, 8, 1007-1032, (2008) · Zbl 1170.35082 [23] Ding, Y. H.; 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 [24] Kryszewski, W.; Szulkin, A., Generalized linking theorem with an application to semilinear Schrödinger equations, Advances in Differential Equations, 3, 3, 441-472, (1998) · Zbl 0947.35061 [25] Szulkin, A.; Zou, W., Homoclinic orbits for asymptotically linear Hamiltonian systems, Journal of Functional Analysis, 187, 1, 25-41, (2001) · Zbl 0984.37072 [26] 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 [27] Willem, M., Minimax Theorems. Minimax Theorems, Progress in Nonlinear Differential Equations and their Applications, 24, (1996), Boston, Mass, USA: Birkhäuser, Boston, Mass, USA [28] Ackermann, N., A superposition principle and multibump solutions of periodic Schrödinger equations, Journal of Functional Analysis, 234, 2, 277-320, (2006) · Zbl 1126.35057
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