Regularity criteria of Leray-Hopf weak solutions to the three-dimensional Navier-Stokes equations in some critical spaces such as Lorentz space, Morrey space and multiplier space are derived in terms of two partial derivatives, 1 u 1 , 2 u 2 , of velocity fields.
In this work, we investigate the following fourth-order delay differential equation of boundary value problem with p-Laplacian(Φp(u))(t) + a(t)f(t, u(t- τ), u(t)) = 0, 0 < t < 1;u(0) = u(0) = 0, u(1) = αu(η);u(t) = 0,- τ≤ t ≤ 0.By using Schauder fixed-point theorem, some sufficient conditions are obtained which guarantee the fourth-order delay differential equation of boundary value problem with p-Laplacian has at least one positive solution. Some corresponding examples are presented to illustrate the application of our main results.
By using cone expansion-compression theorem in this paper, we study boundary value problems for a coupled system of nonlinear third-order differential equation. Some sufficient conditions are obtained which guarantee the boundary value problems for a coupled system of nonlinear third-order differential equation has at least one positive solution. Some examples are given to verify our results.
