Existence of fast homoclinic solutions for a class of second-order damped vibration systems

摘要

By applying the mountain pass theorem in critical point theory, the existence of fast homoclinic solutions is obtained for the following second-order damped vibration system: $$ddot{u}(t)+q(t)dot{u}(t)-L(t)u(t)-a(t) iglert u(t) igrert ^{p-2}u(t)+abla Wigl(t,u(t)igr)=0, $$ where (pin(2,+infty)), (tin{ mathbb {R}}), (uin{ mathbb {R}}^{N}), (L(t)) is a positive definite symmetric matrix-valued function for all (tin{ mathbb {R}}), (Win C^{1}({mathbb {R}}imes{ mathbb {R}}^{N},{mathbb {R}})) is not periodic in t, (a(t)) is a continuous, positive function on ({mathbb {R}}) and (q:{mathbb {R}}ightarrow{ mathbb {R}}) is a continuous function and (Q(t)=int_{0}^{t}q(s),ds) with (lim_{ ert t ert ightarrow+infty}Q(t)=+infty).KeywordsFast homoclinic solutions??Mountain pass theorem??Existence??Critical point??MSC34C37??35A15??37J45??47J30??1 IntroductionConsider fast homoclinic solutions of the following second-order system: $$ ddot{u}(t)+q(t)dot {u}(t)-L(t)u(t)-a(t) iglert u(t) igrert ^{p-2}u(t)+abla Wigl(t,u(t)igr)=0, $$ (1.1) where (pin(2,+infty)), (tin{ mathbb {R}}), (uin{ mathbb {R}}^{N}), (L(t)) is a positive definite symmetric matrix-valued function for all (tin{ mathbb {R}}), (Win C^{1}({mathbb {R}}imes{ mathbb {R}}^{N},{mathbb {R}})) is not periodic in t, (a(t)) is a continuous, positive function on ({mathbb {R}}), and (q:{mathbb {R}}ightarrow{ mathbb {R}}) is a continuous function and (Q(t)=int_{0}^{t}q(s),ds) with $$ lim_{ ert t ert ightarrow+infty}Q(t)=+infty. $$ (1.2)When (q(t)equiv0) and (L(t)equiv0), problem (1.1) reduces to the following special second-order Hamiltonian system: $$ ddot{u}(t)-a(t) iglert u(t) igrert ^{p-2}u(t)+abla Wigl(t,u(t)igr)=0, quadmbox{a.e. }t in{ mathbb {R}}. $$ (1.3)In [1, 2], and [3], the authors considered homoclinic solutions for the special Hamiltonian system (1.3) in a weighted Sobolev space and obtained some results by using the mountain pass theorem in critical point theory. For the applications of mountain pass theorem, please see the references [4] and [5]. In [6], Benci and Fortunato investigated a class of nonlinear Dirichlet problems in a weighted Sobolev space.