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Homoclinic orbits for asymptotically linear Hamiltonian systems. (English) Zbl 0984.37072

The existence of a homoclinic orbit is proved in the paper for a Hamiltonian system \[ \dot z=JH_z(z,t),\tag{1} \] where \(z=(p,q)\in \mathbb R^{2N}\) and \(J=\left (\begin{smallmatrix} 0 & -I\\ I & 0\end{smallmatrix} \right)\). Furthermore, \(H(z,t)=\frac{1}{2}Az\cdot z+G(z,t)\) and \(H(0,t)=0\) with \(G_z(z,t)/|z|\to 0\) uniformly in \(t\) as \(z\to 0\), \(G\) is 1-periodic in \(t\) and asymptotically linear at infinity, and \(JA\) is a hyperbolic matrix. \(G\) has additional properties. A variational method is used to get an abstract theorem which is applied for showing a homoclinic orbit of (1). That theorem is also used to show a decaying solution of an asymptotically linear Schrödinger equation \(-\triangle u+V(x)u=f(x,u)\) for \(x\in \mathbb R^N\), \(V\in C(\mathbb R^n,\mathbb R)\) and \(f\in C(\mathbb R^N,\mathbb R)\).

MSC:

37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010)
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References:

[1] Ambrosetti, A; Zelati, V.Coti, Multiplicité des orbites homoclines pour des systèmes conservatifs, C.R. acad. sci. Paris, 314, 601-604, (1992) · Zbl 0780.49008
[2] Carrião, P.C; Miyagaki, O.H, Existence of homoclinic solutions for a class of time-dependent Hamiltonian systems, J. math. anal. appl., 230, 157-172, (1999) · Zbl 0919.34046
[3] Zelati, V.Coti; Ekeland, I; Séré, E, A variational approach to homoclinic orbits in Hamiltonian systems, Math. ann., 228, 133-160, (1990) · Zbl 0731.34050
[4] Zelati, V.Coti; Rabinowitz, P.H, Homoclinic orbits for second order Hamiltonian systems possessing superquadratic potentials, J. amer. math. soc., 4, 693-727, (1991) · Zbl 0744.34045
[5] Ding, Y, Existence and multiciplicity results for homoclinic solutions to a class of Hamiltonian systems, Nonlinear anal., 25, 1095-1113, (1995) · Zbl 0840.34044
[6] Ding, Y; Girardi, M, Periodic and homoclinic solutions to a class of Hamiltonian systems with the potentials changing sign, Dynam. system appl., 2, 131-145, (1993) · Zbl 0771.34031
[7] Ding, Y; Li, S, Homoclinic orbits for first order Hamiltonian systems, J. math. anal. appl., 189, 585-601, (1995) · Zbl 0818.34023
[8] Ding, Y; Willem, M, Homoclinic orbits of a Hamiltonian system, Z. angew. math. phys., 50, 759-778, (1999) · Zbl 0997.37041
[9] Hofer, H; Wysocki, K, First order elliptic systems and the existence of homoclinic orbits in Hamiltonian systems, Math. ann., 228, 483-503, (1990) · Zbl 0702.34039
[10] Jeanjean, L, On the existence of bounded palais – smale sequences and application to a landesman – lazer type problem set on RN, Proc. roy. soc. Edinburgh, 129A, 787-809, (1999) · Zbl 0935.35044
[11] Kryszewski, W; Szulkin, A, Generalized linking theorem with an application to semilinear Schrödinger equation, Adv. differential equations, 3, 441-472, (1998) · Zbl 0947.35061
[12] Lions, PL, The concentration-compactness principle in the calculus of variations. the locally compact case. part II, Ann. inst. H. Poincaré, anal. non linéaire, 1, 223-283, (1984) · Zbl 0704.49004
[13] Omana, W; Willem, M, Homoclinic orbits for a class of Hamiltonian systems, Differential integral equations, 5, 1115-1120, (1992) · Zbl 0759.58018
[14] Rabinowitz, P.H, Homoclinic orbits for a class of Hamiltonian systems, Proc. roy. soc. Edinburgh, 114, 33-38, (1990) · Zbl 0705.34054
[15] Rabinowitz, P.H; Tanaka, K, Some results on connecting orbits for a class of Hamiltonian systems, Math. Z., 206, 472-499, (1991) · Zbl 0707.58022
[16] Reed, M; Simon, B, Methods of modern mathematical physics, (1978), Academic Press New York
[17] Séré, E, Existence of infinitely many homoclinic orbits in Hamiltonian systems, Math. Z., 209, 27-42, (1992) · Zbl 0725.58017
[18] Séré, E, Looking for the Bernoulli shift, Ann. inst. H. Poincaré, anal. non linéaire, 10, 561-590, (1993) · Zbl 0803.58013
[19] Stuart, C.A, Bifurcation into spectral gaps, Bull. belg. math. soc., 59, (1995) · Zbl 0864.47037
[20] Tanaka, K, Homoclinic orbits in a first order superquadratic Hamiltonian system: convergence of subharmonic orbits, J. differential equations, 94, 315-339, (1991) · Zbl 0787.34041
[21] Willem, M, Minimax theorems, (1996), Birkhäuser Boston
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