Abstract
In this paper, we consider a class of non-periodic damped vibration problems with superquadratic nonlinearities. We study the existence of nontrivial ground state homoclinic orbits for this class of damped vibration problems under some conditions weaker than those previously assumed. To the best of our knowledge, there has been no work focused on this case. MSC: 49J40, 70H05.
Highlights
1 Introduction and main results We shall study the existence of ground state homoclinic orbits for the following nonperiodic damped vibration system: u (t) + Mu (t) – L(t)u(t) + Hu t, u(t) =, t ∈ R, ( . )
We say that a solution u(t) of ( . ) is homoclinic if u(t) ∈ C (R, RN ) such that u(t) → and u (t) → as |t| → ∞
We introduce respectively on E the following new inner product and norm: u, v := |χ | / u, |χ | / v, u = u, u /, ( . )
Summary
With respect to the inner product ·, · E. To see (c), if un → |·|ω u and Iλ(un) ≥ c, u+n → u+ and u–n u– in E, un → u a.e. on R, going to a subsequence if necessary It follows from the weak lower semicontinuity of the norm, Fatou’s lemma and the fact H(t, u) + W (t) ≥ for all t ∈ R and u ∈ RN by We still need to verify condition (d), that is, the following two lemmas. Proof Suppose by contradiction that there exist Rn → ∞, λn ∈ [ , λ ] and un = vn + snz ∈ ∂QR such that Iλn (un) >. Proof Let {un} be the sequence obtained in Lemma.
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
