Electronic Journal of Differential Equations,
Vol. 2014 (2014), No. 111, pp. 1-10.
Title: Existence and multiplicity of homoclinic solutions for
p(t)-Laplacian systems with subquadratic potentials
Authors: Bin Qin (China Three Gorges Univ., Yichang, Hubei, China)
Peng Chen (China Three Gorges Univ., Yichang, Hubei, China)
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.