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HOMOCLINIC SOLUTIONS FOR SUBQUADRATIC HAMILTONIAN SYSTEMS WITHOUT COERCIVE CONDITIONS
- Source :
- Taiwanese J. Math. 18, no. 4 (2014), 1089-1105
- Publication Year :
- 2014
- Publisher :
- Mathematical Society of the Republic of China, 2014.
-
Abstract
- In this paper we investigate the existence and multiplicity of classical homoclinic solutions for the following second order Hamiltonian systems $$ (\mbox{HS}) \ddot u-L(t)u+\nabla W(t,u)=0, $$ where $L\in C(\mathbb R,\mathbb R^{n^2})$ is a symmetric and positive definite matrix for all $t\in \mathbb R$, $W\in C^1(\mathbb R\times\mathbb R^n,\mathbb R)$ and $\nabla W(t,u)$ is the gradient of $W$ at $u$. The novelty of this paper is that, assuming that $L$ is bounded in the sense that there are constants $0\lt \tau_1\lt \tau_2$ such that $\tau_1 |u|^2\leq (L(t)u,u)\leq \tau_2 |u|^2$ for all $(t,u)\in \mathbb R\times \mathbb R^n$ and $W(t,u)$ is of subquadratic growth at infinity, we are able to establish two new criteria to guarantee the existence and multiplicity of classical homoclinic solutions for (HS), respectively. Recent results in the literature are extended and significantly improved.
- Subjects :
- Discrete mathematics
homoclinic solutions
General Mathematics
variational methods
Order (ring theory)
critical point
34C37
Multiplicity (mathematics)
35B38
Positive-definite matrix
genus
Critical point (mathematics)
Hamiltonian system
35A15
Combinatorics
Bounded function
Nabla symbol
Homoclinic orbit
Mathematics
Subjects
Details
- Language :
- English
- Database :
- OpenAIRE
- Journal :
- Taiwanese J. Math. 18, no. 4 (2014), 1089-1105
- Accession number :
- edsair.doi.dedup.....44ca38176ec3a34669117e09c334d9af