In this paper, we are concerned with the following critical fractional Choquard equation$ (-\Delta)^{s}u + V(x)u = (\mathcal{K}_{\mu}*|u|^{2^{*}_{\mu,s}})|u|^{2^{*}_{\mu,s}-2}u, \ u \in D^{s,2}(\mathbb{R}^{N}), $where $ s\in (0,1) $, N\geq 3 \max\{N-4s, 0\}<\mu<N 2^{*}_{\mu,s} \frac{2N-\mu}{N-2s} V(x) is a potential function, and \mathcal{K}_{\mu} Riesz potential. first part of combining ...

We introduce the class of “right almost V-rings” which is properly between the classes of right V-rings and right good rings. A ring R is called a right almost V-ring if every simple R-module is almost injective. It is proved that R is a right almost V-ring if and only if for every R-module M, any complement of every simple submodule of M is a direct summand. Moreover, R is a right almost V-rin...

In this paper our focus is to study certain covering properties in topological spaces by using semi-open covers. A part of this article deals with Rothberger-type covering properties. The notions of s-Rothberger, almost s-Rothberger, star s-Rothberger, almost star s-Rothberger, strongly star s-Rothberger spaces are defined and corresponding properties are investigated.

A new and simpler proof is given of the result of P. Rabinowitz for nontrivial time periodic solutions of a vibrating string equation utt-ux, + g(u) = 0 and Dirichlet boundary conditions on a finite interval. We assume essentially that g is nondecreasing, and g(u)/u+m as Iu(+w. The proof uses a modified form (PS), of the Palais-Smale condition (PS). Let g : R-+R be a continuous nondecreasing fu...