In this paper, we discuss the existence and non-existence of weak solutions to non-linear problem with a fractional p-Laplacian introduced by ?-Hilfer operator, combining technique Nehari manifolds fibering maps. Also, obtain some results on operator manifold via Euler functional.

We establish the topological s-cobordism surgery sequence for any closed oriented 4-manifold X homotopy equivalent to a connected sum X1# · · ·#Xn such that each topological 4-manifold Xi has fundamental group Γi of class SA. As a corollary, given Γi in a subclass ECS, if Γi is torsionfree and π2(Xi)⊗ Z2 = 0 (e.g. Xi aspherical), then X is topologically s-rigid. An application is an s-fibering ...

Let M be a closed 4-manifold with a free circle action. If the orbit manifold N3 satisfies an appropriate fibering condition, then we show how to represent a cone in H2(M ;R) by symplectic forms. This generalizes earlier constructions by Thurston, Bouyakoub and Fernández et al . In the case that M is the product 4-manifold S1 ×N , our construction complements our previous results and allows us ...

We prove that cubulated hyperbolic groups are virtually special. The proof relies on results of Haglund and Wise which also imply that they are linear groups, and quasi-convex subgroups are separable. A consequence is that closed hyperbolic 3-manifolds have finite-sheeted Haken covers, which resolves the virtual Haken question of Waldhausen and Thurston’s virtual fibering question. An appendix ...

In this note, we compute the virtual first Betti numbers of 4-manifolds fibering over S1 with prime fiber. As an application, we show that if such a manifold is symplectic with nonpositive Kodaira dimension, then the fiber itself is a sphere or torus bundle over S1. In a different direction, we prove that if the 3-dimensional fiber of such a 4-manifold is virtually fibered then the 4-manifold i...