English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Zendy Open

Presents the pdf analysis as premium feature

PDF Analysis

Insights

Presents the summarisation as premium feature

Summarisation

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the zaia usage as premium feature

ZAIA

Zendy Tools

Zendy Plus

Presents the access of premium content as premium feature

Premium Content

We are concerned with the existence of homoclinic solutions for the following second order nonautonomous singular Hamiltonian systems u ¨ + a t W u u = 0 , (HS) where - ∞ < t < + ∞ , u = u 1 , u 2 ,   … , u N ∈ ℝ N N ≥ 3 , a : ℝ → ℝ is a continuous bounded function, and the potential W : ℝ N ∖ { ξ } → ℝ has a singularity at 0 ≠ ξ ∈ ℝ N , and W u u is the gradient of W at u . The novelty of this paper is that, for the case that N ≥ 3 and (HS) is nonautonomous (neither periodic nor almost periodic), we show that (HS) possesses at least one nontrivial homoclinic solution. Our main hypotheses are the strong force condition of Gordon and the uniqueness of a global maximum of W . Different from the cases that (HS) is autonomous a t ≡ 1 or (HS) is periodic or almost periodic, as far as we know, this is the first result concerning the case that (HS) is nonautonomous and N ≥ 3 . Besides the usual conditions on W , we need the assumption that a ′ t < 0 for all t ∈ ℝ to guarantee the existence of homoclinic solution. Recent results in the literature are generalized and significantly improved.

Homoclinic Solutions for a Class of Second Order Nonautonomous Singular Hamiltonian Systems