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Existence of infinitely many homoclinic orbits in Hamiltonian systems. (English) Zbl 1237.37044

In this paper the authors consider the following second-oder Hamiltonian system

$\stackrel{¨}{u}-L\left(t\right)u\left(t\right)+\nabla W\left(t,u\left(t\right)\right)=0\phantom{\rule{2.em}{0ex}}\left(1\right)$

where $t\in ℝ,u\in {ℝ}^{N}$, $L:ℝ\to {ℝ}^{N×N}$ and $W:ℝ×{ℝ}^{N}\to ℝ$. A nonzero solution $u$ of (1) is said to be homoclinic (to 0) if $u\left(t\right)\to 0$ as $|t|\to \infty$.

The authors deal with the existence and multiplicity of homoclinic solutions under the assumption that $W\left(t,x\right)$ is indefinite sign and subquadratic as $|x|\to \infty$ by using the genus properties in critical theory. The results in this paper extends the results in [Z. Zhang and R. Yuan, Nonlinear Anal., Theory Methods Appl. 71, No. 9, A, 4125–4130 (2009; Zbl 1173.34330)] and [J. Sun, H. Chen and J. J. Nieto, J. Math. Anal. Appl. 373, No. 1, 20–29 (2011; Zbl 1230.37079)]. The main results are the following.

Theorem 1.1 Assume that $L$ and $W$ satisfy the following conditions:

$L\in C\left(ℝ,{ℝ}^{N×N}\right)$ is a definite symmetric matrix for all $t\in ℝ$ and there exists a constant $\beta >0$ such that

$\left(L\left(t\right)x,x\right)\ge {\beta |x|}^{2}$

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

There exist two constants $1<{r}_{1}<{r}_{2}<2$ and two functions ${a}_{1},{a}_{2}\in {L}^{\frac{2}{2-{r}_{1}}}\left(ℝ,\left[0,+\infty \right)\right)$ such that

$|W\left(t,x\right)|\le {a}_{1}\left(t\right){|x|}^{{r}_{1}}$

for all $\left(t,x\right)\in ℝ×{ℝ}^{N},|x|\le 1$, and

$|W\left(t,x\right)|\le {a}_{2}\left(t\right){|x|}^{{r}_{2}}$

for all $\left(t,x\right)\in ℝ×{ℝ}^{N},|x|\ge 1$;

There exist two functions $b\in {L}^{2/\left(2-{r}_{1}\right)}\left(ℝ,\left[0,+\infty \right)\right)$ and $\phi \in C\left(\left[0,+\infty \right),\left[0,+\infty \right)\right)$ such that

$|\nabla W\left(t,x\right)|\le b\left(t\right)\phi \left(|x|\right)$

for all $\left(t,x\right)\in ℝ×{ℝ}^{N}$, where $\phi \left(s\right)=O\left({s}^{{r}_{1}-1}\right)$ as $s\to {0}^{+}$;

There exist an open set $J\subset ℝ$ and two constants ${r}_{3}\in \left(1,2\right)$ and $\eta >0$ such that

$W\left(t,x\right)\ge {\eta |x|}^{{r}_{3}}$

for all $\left(t,x\right)\in J×{ℝ}^{N}$, $|x|\le 1$.

Then Problem (1) has at least one nontrivial homoclinic orbit.

Theorem 1.2 Assume that $L$ and $W$ satisfy (L), (${W}_{1}$), (${W}_{2}$), (${W}_{3}$) and that $W\left(t,-x\right)=W\left(t,x\right)$ for all $t\in ℝ$ and $x\in {ℝ}^{N}$. Then Problem (1) has infinitely many homoclinic orbits.

##### MSC:
 37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods 34C37 Homoclinic and heteroclinic solutions of ODE