Electron. J. Diff. Equ., Vol. 2014 (2014), No. 111, pp. 1-10.

Existence and multiplicity of homoclinic solutions for p(t)-Laplacian systems with subquadratic potentials

Bin Qin, Peng Chen

Abstract:
By using the genus properties, we establish some criteria for the second-order p(t)-Laplacian system
$$ \frac{d}{dt}\big(|\dot{u}(t)|^{p(t)-2}\dot{u}(t)\big)-a(t)|u(t)|^{p(t)-2}u(t) +\nabla W(t, u(t))=0 $$
to have at least one, and infinitely many homoclinic orbits. where $t\in {\mathbb{R}},\; u\in {\mathbb{R}}^{N}$, $p(t)\in C(\mathbb{R},\mathbb{R})$ and $p(t)>1$, $a\in C({\mathbb{R}}, {\mathbb{R}})$ and $W\in C^{1}({\mathbb{R}}\times {\mathbb{R}}^{N}, {\mathbb{R}})$ may not be periodic in t.

Submitted December 17, 2013. Published April 16, 2014.
Math Subject Classifications: 34C37, 58E05, 70H05.
Key Words: Homoclinic solutions; p(t)-Laplacian systems; genus.

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Bin Qin
College of Science, China Three Gorges University
Yichang, Hubei 443002, China
email: 1070751409@qq.com
Peng Chen
College of Science, China Three Gorges University
Yichang, Hubei 443002, China
email: pengchen729@sina.com

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