An analytic construction of singular solutions related to a critical Yamabe problem

We answer affirmatively a question of Aviles posed in 1983, concerning the construction of singular solutions of semilinear equations without using phase-plane analysis. Fully exploiting the semilinearity and the stability of the linearized operator in any dimension, our techniques involve a careful gluing in weighted $L^\infty$ spaces that handles multiple occurrences of criticality, without the need of derivative estimates. The above solution constitutes an \emph{Ansatz} for the Yamabe problem with a prescribed singular set of maximal dimension $(n-2)/2$, for which, using the same machinery, we provide an alternative construction to the one given by Pacard. His linear theory uses $L^p$-theory on manifolds, while our approach studies the equations in the ambient space and is therefore suitable for generalization to nonlocal problems. In a forthcoming paper, we will prove analogous results in the fractional setting.Comment: 24 page

Hazardous Apoptotic Effects of 2-Bromopropane on Maturation of Mouse Oocytes, Fertilization, and Fetal Development

2-Bromopropane (2-BP) is used as an alternative to ozone-depleting cleaning solvents. Previously, we reported that 2-BP has cytotoxic effects on mouse blastocysts and is associated with defects in subsequent development. Here, we further investigate the effects of 2-BP on oocyte maturation and subsequent pre- and post-implantation development, both in vitro and in vivo. Notably, 2-BP induced a significant reduction in the rates of oocyte maturation, fertilization, and in vitro embryonic development. Treatment of oocytes with 2-BP during in vitro maturation (IVM) resulted in increased resorption of postimplantation embryos and decreased fetal weights. Experiments with a mouse model disclosed that consumption of drinking water containing 20 μM 2-BP led to decreased oocyte maturation in vivo and fertilization in vitro, as well as impairment of early embryonic development. Interestingly, pretreatment with a caspase-3-specific inhibitor effectively prevented 2-BP-triggered hazardous effects, suggesting that embryonic impairment by 2-BP occurs via a caspase-dependent apoptotic process. A study using embryonic stem cells as the assay model conclusively demonstrated that 2-BP induces cell death processes through apoptosis and not necrosis, and inhibits early embryo development in mouse embryonic stem cells. These results collectively confirm the hazardous effects of 2-BP on embryos derived from pretreated oocytes