We study the existence of positive solutions for a class of double phase Dirichlet equations which have the combined effects of a singular term and of a parametric superlinear term. The differential operator of the equation is the sum of a $p$-Laplac
ian and of a weighted $q$-Laplacian ($q<p$) with discontinuous weight. Using the Nehari method, we show that for all small values of the parameter $lambda>0$, the equation has at least two positive solutions.
In this survey we present applications of the ideas of complement and neighborhood in the theory embeddings of manifolds into Euclidean space (in codimension at least three). We describe how the combination of these ideas gives a reduction of embedda
bility and isotopy problems to algebraic problems. We present a more clarified exposition of the Browder-Levine theorem on realization of normal systems. Most of the survey is accessible to non-specialists in the theory of embeddings.
We investigate the classical Alexandroff-Borsuk problem in the category of non-triangulable manifolds: Given an $n$-dimensional compact non-triangulable manifold $M^n$ and $varepsilon > 0$, does there exist an $varepsilon$-map of $M^n$ onto an $n$-di
mensional finite polyhedron which induces a homotopy equivalence?
The main results of this paper are: (1) If a space $X$ can be embedded as a cellular subspace of $mathbb{R}^n$ then $X$ admits arbitrary fine open coverings whose nerves are homeomorphic to the $n$-dimensional cube $mathbb{D}^n$; (2) Every $n$-dimens
ional cell-like compactum can be embedded into $(2n+1)$-dimensional Euclidean space as a cellular subset; and (3) There exists a locally compact planar set which is acyclic with respect to v{C}ech homology and whose fine coverings are all nonacyclic.
It has been known for a long time that the fundamental group of the quotient of $RR ^3$ by the Case-Chamberlin continuum is nontrivial. In the present paper we prove that this group is in fact, uncountable.
