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Existence of two almost homoclinic solutions for p(t)-Laplacian Hamiltonian systems with a small perturbation

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Abstract

In this paper we study the existence of two almost homoclinic solutions for the following second order p(t)-Laplacian Hamiltonian systems with a small perturbation

$$\begin{aligned} \frac{d}{dt}\big (|\dot{u}(t)|^{p(t)-2} \dot{u}(t)\big )-a(t)|u(t)|^{p(t)-2}u(t)+\nabla W(t,u(t))=f(t), \end{aligned}$$

where \(t\in {\mathbb {R}}\), \(u\in {\mathbb {R}}^n\), \(p\in C({\mathbb {R}},{\mathbb {R}})\) with \(p(t)>1\), \(a\in C({\mathbb {R}},{\mathbb {R}})\), \(W\in C^1({\mathbb {R}}\times {\mathbb {R}}^n,{\mathbb {R}})\) and \(\nabla W(t,u)\) is the gradient of W(tu) at u, \(f\in C({\mathbb {R}},{\mathbb {R}}^n)\) and belongs to \(L^{q(t)}({\mathbb {R}},{\mathbb {R}}^n)\). The point is that, assuming that a(t) is bounded in the sense that there are two constants \(0<\tau _1<\tau _2<\infty \) such that \(\tau _1\le a(t)\le \tau _2 \) for all \(t \in {\mathbb {R}}\), W(tu) is of super-p(t) growth as \(|u|\rightarrow \infty \) and satisfies some other reasonable hypothesis, f is sufficiently small in \(L^{q(t)}({\mathbb {R}},{\mathbb {R}}^n)\), we provide one new criterion to ensure the existence of two almost homoclinic solutions. Recent results in the literature are extended and significantly improved.

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Acknowledgments

This work is supported by the National Natural Science Foundation of China (Grant Nos.11101304,11031002,11371058), RFDP (Grant No. 20110003110004), and the Grant of Beijing Education Committee Key Project (Grant No. KZ20130028031).

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Correspondence to Ziheng Zhang.

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Zhang, Z., Yuan, R. Existence of two almost homoclinic solutions for p(t)-Laplacian Hamiltonian systems with a small perturbation. J. Appl. Math. Comput. 52, 173–189 (2016). https://doi.org/10.1007/s12190-015-0936-0

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