摘要：In this paper, we study the existence of nontrivial homoclinic orbits of a dynamic equation on time scales T $\mathbb{T}$ of the form { ( p ( t ) u Δ ( t ) ) Δ + q σ ( t ) u σ ( t ) = f ( σ ( t ) , u σ ( t ) ) , △ -a.e. t ∈ T , u ( ± ∞ ) = u Δ ( ± ∞ ) = 0 . $$ \left \{ \textstyle\begin{array}{l} ( p(t)u^{\Delta}(t) ) ^{\Delta}+q^{\sigma}(t)u^{\sigma}(t)= f(\sigma(t),u^{\sigma}(t)),\quad \triangle\text{-a.e. } t\in\mathbb{T}, \\ u(\pm\infty)=u^{\Delta}(\pm\infty)=0. \end{array}\displaystyle \right . $$ We construct a variational framework of the above-mentioned problem, and some new results on the existence of a homoclinic orbit or an unbounded sequence of homoclinic orbits are obtained by using the mountain pass lemma and the symmetric mountain pass lemma, respectively. The interesting thing is that the variational method and the critical point theory are used in this paper. It is notable that in our study any periodicity assumptions on p ( t ) $p(t)$ , q ( t ) $q(t)$ and f ( t , u ) $f(t,u)$ are not required.
