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Fast homoclinic solutions for a class of damped vibration problems with subquadratic potentials
Author(s) -
Chen Peng,
Tang X. H.
Publication year - 2013
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201100287
Subject(s) - homoclinic orbit , multiplicity (mathematics) , mathematics , mountain pass theorem , critical point (mathematics) , vibration , class (philosophy) , mathematical analysis , function (biology) , pure mathematics , bifurcation , nonlinear system , physics , computer science , acoustics , quantum mechanics , artificial intelligence , evolutionary biology , biology
In this paper, we deal with the existence and multiplicity of homoclinic solutions of the following damped vibration problems\documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty} \begin{eqnarray*} \ddot{u}(t)+q(t)\dot{u}(t)-L(t)u(t)+\nabla W(t, u(t))=0, \end{eqnarray*} \end{document} where L ( t ) and W ( t , x ) are neither autonomous nor periodic in t . Our approach is variational and it is based on the critical point theory. We prove existence and multiplicity results of fast homoclinic solutions under general growth conditions on the potential function. Our theorems appear to be the first such result and our results extend some recent works.