Existence of two almost homoclinic solutions for p(t)-Laplacian Hamiltonian systems with a small perturbation

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• 作者：Ziheng Zhang ; Rong Yuan
• 关键词：Homoclinic solutions ; Critical point ; Variational methods ; Mountain pass theorem
• 刊名：Journal of Applied Mathematics and Computing
• 出版年：2016
• 出版时间：October 2016
• 年：2016
• 卷：52
• 期：1-2
• 页码：173-189
• 全文大小：480 KB
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Computational Mathematics and Numerical Analysis
  Applied Mathematics and Computational Methods of Engineering
  Theory of Computation
  Mathematics of Computing
  
• 出版者：Springer Berlin / Heidelberg
• ISSN：1865-2085
• 卷排序：52

文摘

In this paper we study the existence of two almost homoclinic solutions for the following second order p(t)-Laplacian Hamiltonian systems with a small perturbation $$\begin{aligned} \frac{d}{dt}\big (|\dot{u}(t)|^{p(t)-2} \dot{u}(t)\big )-a(t)|u(t)|^{p(t)-2}u(t)+\nabla W(t,u(t))=f(t), \end{aligned}$$where \(t\in {\mathbb {R}}\), \(u\in {\mathbb {R}}^n\), \(p\in C({\mathbb {R}},{\mathbb {R}})\) with \(p(t)>1\), \(a\in C({\mathbb {R}},{\mathbb {R}})\), \(W\in C^1({\mathbb {R}}\times {\mathbb {R}}^n,{\mathbb {R}})\) and \(\nabla W(t,u)\) is the gradient of W(t, u) at u, \(f\in C({\mathbb {R}},{\mathbb {R}}^n)\) and belongs to \(L^{q(t)}({\mathbb {R}},{\mathbb {R}}^n)\). The point is that, assuming that a(t) is bounded in the sense that there are two constants \(0<\tau _1<\tau _2<\infty \) such that \(\tau _1\le a(t)\le \tau _2 \) for all \(t \in {\mathbb {R}}\), W(t, u) is of super-p(t) growth as \(|u|\rightarrow \infty \) and satisfies some other reasonable hypothesis, f is sufficiently small in \(L^{q(t)}({\mathbb {R}},{\mathbb {R}}^n)\), we provide one new criterion to ensure the existence of two almost homoclinic solutions. Recent results in the literature are extended and significantly improved.KeywordsHomoclinic solutionsCritical pointVariational methodsMountain pass theoremMathematics Subject Classification34C3735A1535B38References1.Admas, R.A.: Sobolev Spaces. Academic Press, New York (1975)Google Scholar2.Alves, C.O., Carrião, P.C., Miyagaki, O.H.: Existence of homoclinic orbits for asymptotically periodic systems involving Duffing-like equation. Appl. Math. 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Lett. 16(8), 1283–1287 (2003)MathSciNetCrossRefMATHGoogle ScholarCopyright information© Korean Society for Computational and Applied Mathematics 2015Authors and AffiliationsZiheng Zhang1Email authorRong Yuan21.Department of MathematicsTianjin Polytechnic UniversityTianjinChina2.Department of Mathematical SciencesBeijing Normal UniversityBeijingChina About this article CrossMark Print ISSN 1598-5865 Online ISSN 1865-2085 Publisher Name Springer Berlin Heidelberg About this journal Reprints and Permissions Article actions function trackAddToCart() { var buyBoxPixel = new webtrekkV3({ trackDomain: "springergmbh01.webtrekk.net", trackId: "196033507532344", domain: "link.springer.com", contentId: "springer_com.buybox", product: "10.1007/s12190-015-0936-0_Existence of two almost homoclinic", productStatus: "add", productCategory : { 1 : "ppv" }, customEcommerceParameter : { 9 : "link.springer.com" } }); buyBoxPixel.sendinfo(); } function trackSubscription() { var subscription = new webtrekkV3({ trackDomain: "springergmbh01.webtrekk.net", trackId: "196033507532344", domain: "link.springer.com", contentId: "springer_com.buybox" }); subscription.sendinfo({linkId: "inst. subscription info"}); } window.addEventListener("load", function(event) { var viewPage = new webtrekkV3({ trackDomain: "springergmbh01.webtrekk.net", trackId: "196033507532344", domain: "link.springer.com", contentId: "SL-article", product: "10.1007/s12190-015-0936-0_Existence of two almost homoclinic", productStatus: "view", productCategory : { 1 : "ppv" }, customEcommerceParameter : { 9 : "link.springer.com" } }); viewPage.sendinfo(); }); Log in to check your access to this article Buy (PDF)EUR 34,95 Unlimited access to full article Instant download (PDF) Price includes local sales tax if applicable Find out about institutional subscriptions Export citation .RIS Papers Reference Manager RefWorks Zotero .ENW EndNote .BIB BibTeX JabRef Mendeley Share article Email Facebook Twitter LinkedIn Cookies We use cookies to improve your experience with our site. 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