In this article we study the positivity of the 4-th order Paneitz operator for closed 3-manifolds. We prove that the connected sum of two such 3-manifold retains the same positivity property. We also solve the analogue of the Yamabe equation for such a manifold.
The aim of this paper is to show that Liouville type property is a sufficient and necessary condition for the regularity of weak solutions of nonlinear elliptic systems of the higher order.
The paper deals with the existence of time-periodic solutions to the beam equation, in which terms expressing torsion and damping are also considered. The existence of periodic solutions is proved in the cas of time-periodic outer forces by means of an apriori estimate and the Fourier method.
In this article, we prove the partial exact controllability of a nonlinear system. We use semigroup formulation together with fixed point approach to study the nonlinear system.