Fast homoclinic solutions for damped vibration problems with superquadratic potentials

• Published in: Boundary value problems 2018-12, Vol.2018 (1), p.1-14, Article 183
• Main Authors: Zhu, Xinhe, Zhang, Ziheng
• Format: Article
• Language: English
• Subjects: Analysis; Approximations and Expansions; Coercivity; Continuity (mathematics); Critical point; Difference and Functional Equations; Existence theorems; Homoclinic solutions; Mathematical analysis; Mathematics; Mathematics and Statistics; Matrix methods; Mountain pass theorem; Ordinary Differential Equations; Partial Differential Equations; Variational methods; Vibration control
• ISSN: 1687-2770; 1687-2762; 1687-2770
• DOI: 10.1186/s13661-018-1103-y

In this paper we investigate the existence and multiplicity of homoclinic solutions for the following damped vibration problem: DS u ¨ + q ( t ) u ˙ − L ( t ) u + W u ( t , u ) = 0 , where q : R → R is a continuous function, L ∈ C ( R , R n 2 ) is a symmetric and positive definite matrix for all t ∈ R and W ∈ C 1 ( R × R n , R ) . The novelty of this paper is that, assuming lim | t | → + ∞ Q ( t ) = + ∞ ( Q ( t ) = ∫ 0 t q ( s ) d s ) and L is coercive at infinity, we establish one new compact embedding theorem. Subsequently, supposing that W satisfies the global Ambrosetti–Rabinowitz condition, we obtain some new criterion to guarantee the existence of homoclinic solution of ( DS ) using the mountain pass theorem. Moreover, if W is even, then ( DS ) has infinitely many homoclinic solutions. Recent results in the literature are generalized and significantly improved.