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

Qin, Bin ; Chen, Peng

Electronic journal of differential equations, Tome 2014 (2014)

Résumé

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.

Détail

Classification : 34C37, 58E05, 70H05
Keywords: homoclinic solutions, $p(t)$-Laplacian systems, genus