Existence of homoclinic orbits for second order Hamiltonian systems without (AR) condition

Wan, Li-Li

doi:10.7153/dea-01-27

Cited by 6 publications

(8 citation statements)

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“…We point out that (AR) condition, used in all the works mentioned above, implies that V (t, q) has a superquadratic growth as |q| → ∞. Recently, if f = 0, there are some papers dealing with superquadratic potentials V (t, q) no verifying (AR) condition, but satisfying a set of hypotheses different from ours (see e.g., [19] and [20]).…”

Section: Corollary 13mentioning

confidence: 75%

Some multiplicity results of homoclinic solutions for second order Hamiltonian systems

Barile¹,

Salvatore²

2020

Opuscula Math.

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We look for homoclinic solutions \(q:\mathbb{R} \rightarrow \mathbb{R}^N\) to the class of second order Hamiltonian systems \[-\ddot{q} + L(t)q = a(t) \nabla G_1(q) - b(t) \nabla G_2(q) + f(t) \quad t \in \mathbb{R}\] where \(L: \mathbb{R}\rightarrow \mathbb{R}^{N \times N}\) and \(a,b: \mathbb{R}\rightarrow \mathbb{R}\) are positive bounded functions, \(G_1, G_2: \mathbb{R}^N \rightarrow \mathbb{R}\) are positive homogeneous functions and \(f:\mathbb{R}\rightarrow\mathbb{R}^N\). Using variational techniques and the Pohozaev fibering method, we prove the existence of infinitely many solutions if \(f\equiv 0\) and the existence of at least three solutions if \(f\) is not trivial but small enough.

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Section: Corollary 13mentioning

confidence: 75%

Some multiplicity results of homoclinic solutions for second order Hamiltonian systems

Barile¹,

Salvatore²

2020

Opuscula Math.

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“…which is a classical equation which can describe many mechanical systems, such as a pendulum. In the past decades, the existence and multiplicity of periodic solutions and homoclinic orbits for (3) have been studied by many authors via variational methods; see [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17] and the references therein. The periodic assumptions are very important in the study of homoclinic orbits for (3) since periodicity is used to control the lack of compactness due to the fact that (3) is set on all R. Nonperiodic problems are quite different from the ones described in periodic cases.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Recently, by using conditions ( 1 ) and ( 2 ), Chen [22,23] obtained infinitely many nontrivial homoclinic orbits of (1) when satisfies the subquadratic [22] (or superquadratic [23]) growth condition at infinity. In fact, conditions ( 1 ) and ( 2 ) are first used in [14]. As mentioned in [21], there are some matrix-valued functions ( ) satisfying ( 1 ) and ( 2 ) but not satisfying (4).…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Nonperiodic Damped Vibration Systems with Asymptotically Quadratic Terms at Infinity: Infinitely Many Homoclinic Orbits

Chen

2013

Abstract and Applied Analysis

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We study a class of nonperiodic damped vibration systems with asymptotically quadratic terms at infinity. We obtain infinitely many nontrivial homoclinic orbits by a variant fountain theorem developed recently by Zou. To the best of our knowledge, there is no result published concerning the existence (or multiplicity) of nontrivial homoclinic orbits for this class of non-periodic damped vibration systems with asymptotically quadratic terms at infinity.

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“…However, there are lots of potentials which are superquadratic as |u| → +∞ but do not satisfy the (AR) condition. Therefore, many authors have been focusing their attention on obtaining the existence of homoclinic solutions under the conditions weaker than the (AR) condition, see for instance [7,13,14,23,32] and the references listed therein. In addition, to verify the (PS) condition for the corresponding energy functional of (HS), the following coercive assumption on L is often needed:…”

Section: Introductionmentioning

confidence: 99%

Fast homoclinic solutions for damped vibration problems with superquadratic potentials

Zhu

Zhang

2018

Bound Value Probl

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In this paper we investigate the existence and multiplicity of homoclinic solutions for the following damped vibration problem: u + q(t)u-L(t)u + W u (t, u) = 0, (DS) where q : R → R is a continuous function, L ∈ C(R, R n 2) is a symmetric and positive definite matrix for all t ∈ R and W ∈ C 1 (R × R n , R). The novelty of this paper is that, assuming lim |t|→+∞ Q(t) = +∞ (Q(t) = t 0 q(s) ds) and L is coercive at infinity, we establish one new compact embedding theorem. Subsequently, supposing that W satisfies the global Ambrosetti-Rabinowitz condition, we obtain some new criterion to guarantee the existence of homoclinic solution of (DS) using the mountain pass theorem. Moreover, if W is even, then (DS) has infinitely many homoclinic solutions. Recent results in the literature are generalized and significantly improved.

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Existence of homoclinic orbits for second order Hamiltonian systems without (AR) condition

Abstract: Abstract. The existence of homoclinic orbits is obtained for a class of the second order Hamiltonian systemsü(t) − L(t)u(t) + ∇W (t,u(t)) = 0, ∀t ∈ R , by the mountain pass theorem, where W (t,x) needs not to satisfy the global (AR) condition. Mathematics subject classification

Cited by 6 publications

References 0 publications

Some multiplicity results of homoclinic solutions for second order Hamiltonian systems

Some multiplicity results of homoclinic solutions for second order Hamiltonian systems

Nonperiodic Damped Vibration Systems with Asymptotically Quadratic Terms at Infinity: Infinitely Many Homoclinic Orbits

Fast homoclinic solutions for damped vibration problems with superquadratic potentials

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