Some results for impulsive problems via Morse theory

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:409

Article

Agarwal, Ravi P.¹  Bhaskar, T. Gnana²  Perera, Kanishka² 

[1] Texas A&M Univ, Dept Math, Kingsville, TX 78363 USA

[2] Florida Inst Technol, Dept Math Sci, Melbourne, FL 32901 USA

关键词: Impulsive problems;    Asymptotically piecewise linear problems;    Resonance set;    Nontrivial solutions;    Morse theory;    Critical groups;    Saddle point theorem;   

DOI  :  10.1016/j.jmaa.2013.07.036

来源: Elsevier

【 摘 要 】

We use Morse theory to study impulsive problems. First we consider asymptotically piecewise linear problems with superlinear impulses, and prove a new existence result for this class of problems using the saddle point theorem. Next we compute the critical groups at zero when the impulses are asymptotically linear near zero, in particular, we identify an important resonance set for this problem. As an application, we finally obtain a nontrivial solution for asymptotically piecewise linear problems with impulses that are asymptotically linear at zero and superlinear at infinity. Our results here are based on the simple observation that the underlying Sobolev space naturally splits into a certain finite dimensional subspace where all the impulses take place and its orthogonal complement that is free of impulsive effects. (C) 2013 Elsevier Inc. All rights reserved.

【 授权许可】

Free