# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Infinitely many homoclinic orbits for the second order Hamiltonian systems with general potentials. (English) Zbl 1200.37054

The authors consider the second order Hamiltonian system

$\text{(HS)}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}-\stackrel{¨}{u}\left(t\right)+L\left(t\right)u\left(t\right)=\nabla R\left(t,u\right),\phantom{\rule{4pt}{0ex}}\forall t\in ℝ,$

where $L\left(t\right)\in C\left(ℝ,{ℝ}^{{N}^{2}}\right)$ is a $N×N$ symmetric matrix valued function and $R\left(t,u\right)\in {C}^{1}\left(ℝ×{ℝ}^{N},ℝ\right)$. They use the variant fountain theorem to prove that the system (HS) possesses infinitely many homoclinic orbits, assuming that $L\left(t\right)$ satisfies the coercive condition and $R\left(t,u\right)$ satisfies the conditions that

(${R}_{1}$) $R\left(t,u\right)=F\left(t,u\right)+G\left(t,u\right)\phantom{\rule{4pt}{0ex}}\text{and}\phantom{\rule{4pt}{0ex}}F,G\in {C}^{1}\left(ℝ×{ℝ}^{N},ℝ\right)$ are even in $u$;

(${R}_{2}$) There exist $\sigma ,\delta \in \left(1,2\right),{c}_{1}>0,{c}_{2}>0,{c}_{3}>0$ such that

${c}_{1}{|u|}^{\sigma }\le {F}_{u}\left(t,u\right)u\le {c}_{2}{|u|}^{\sigma }+{c}_{3}{|u|}^{\delta }$

for all $\left(t,u\right)\in ℝ×{ℝ}^{N};$

(${R}_{3}$) There exist $\rho \ge 2$ and ${c}_{4}>0$ such that $|{G}_{u}\left(t,u\right)|\le {c}_{4}{\left(1+|u|}^{p-1}\right)$ for all $\left(t,u\right)\in ℝ×{ℝ}^{N}$, moreover, ${lim}_{u\to 0}\frac{{G}_{u}\left(t,u\right)}{|u|}=0$ uniformly for $t\in ℝ$;

(${R}_{4}$) $G\left(t,u\right)\ge 0$ and ${lim}_{|u|\to \infty }\frac{{G}_{u}\left(t,u\right)}{|u|}=+\infty$ uniformly for all $t\in ℝ·$

Conditions (${R}_{2}$)–(${R}_{4}$) imply that $F\left(t,u\right)$ is sub-quadratic and $G\left(t,u\right)$ is super-quadratic, which generalize the sub-quadratic condition that

($R$) $0 where $\gamma \in \left(1,2\right)$.

Reviewer’s remark: In the proof of Lemma 3.3, the authors say that “Set ${w}_{n}:=\frac{{u}_{n}}{\parallel {u}_{n}\parallel }$, then $\parallel {w}_{n}\parallel =1$, ${w}_{n}\to w,\phantom{\rule{4pt}{0ex}}{w}_{n}^{+}\to {w}^{+},\phantom{\rule{4pt}{0ex}}{w}_{n}^{0}\to {w}^{0}$ and ${w}_{n}^{-}\to {w}^{-}$”. It seems that this conclusion is incorrect.

##### MSC:
 37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods 34C37 Homoclinic and heteroclinic solutions of ODE 37C29 Homoclinic and heteroclinic orbits 58E05 Abstract critical point theory