TY - JOUR
AU - Janczewska, Joanna
PY - 2010/04/23
Y2 - 2023/03/28
TI - Homoclinic solutions for a class of autonomous second order Hamiltonian systems with a superquadratic potential
JF - Topological Methods in Nonlinear Analysis
JA - TMNA
VL - 36
IS - 1
SE -
DO -
UR - https://apcz.umk.pl/TMNA/article/view/TMNA.2010.025
SP - 19 - 26
AB - We will prove the existence of a nontrivial homoclinic solutionfor an autonomous second order Hamiltonian system$\ddot{q}+
abla{V}(q)=0$, where $q\in\mathbb{R}^n$, a potential$V\colon\mathbb{R}^n\to\mathbb{R}$ is of the form $V(q)=-K(q)+W(q)$, $K$ and $W$ are $C^{1}$-maps, $K$ satisfies the pinchingcondition, $W$ grows at a superquadratic rate, as $|q|\to\infty$ and $W(q)=o(|q|^2)$, as $|q|\to 0$. A homoclinic solution will be obtained as a weak limitin the Sobolev space $W^{1,2}(\mathbb{R},\mathbb{R}^n)$ of a sequenceof almost critical points of the corresponding action functional. Before passing to a weak limit with a sequence ofalmost critical points each element of this sequence has to beappropriately shifted.
ER -