Abstract
In this paper, a n-dimensional prescribed mean curvature Rayleigh p-Laplacian equation with a deviating argument, $(\varphi_{p}(\frac{u'(t)}{\sqrt{1+|u'(t)|^{2}}}))'+F(t,u'(t))+G(t,u(t-\tau(t)))=e(t)$ , is studied. By means of Mawhin’s continuation theorem and some analysis methods, a new result on the existence of homoclinic solutions for the equation is obtained. Our research enriches the contents of prescribed mean curvature equations.
Highlights
In recent years, the existence of homoclinic solutions has been studied widely for the Hamiltonian systems and the p-Laplacian systems
In [ ], Lzydorek and Janczewska studied the existence of homoclinic solutions for second-order Hamiltonian system in the following form: q + Vq(t, q) = f (t), where q ∈ Rn and V ∈ C (R × Rn, R), V (t, q) = –K(t, q) + W (t, q) is T -periodic with respect to t
Inspired by the above fact, the aim of this paper is to investigate the existence of homoclinic solution to the following n-dimensional prescribed mean curvature equation with a deviating argument: u (t) φp + |u (t)| + F t, u (t) + G t, u t – τ (t) = e(t), ( . )
Summary
The existence of homoclinic solutions has been studied widely for the Hamiltonian systems and the p-Laplacian systems (see [ – ] and the references therein). Feng in [ ] studied the problem of the existence of periodic solution for a prescribed mean curvature Liénard equation u (t). Li in [ ] further studied the existence of periodic solutions for a prescribed mean curvature Rayleigh equation of the form u (t) + f t, u (t) + g t, u t – τ (t) = e(t), and Wang in [ ] discussed the following boundary valued problem:. Inspired by the above fact, the aim of this paper is to investigate the existence of homoclinic solution to the following n-dimensional prescribed mean curvature equation with a deviating argument:. ), F ∈ C(R × Rn; Rn), G ∈ C(R × Rn; Rn), e ∈ C(R; Rn), τ (t) is a continuous T-periodic function and T > is given constant.
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