Multiple solutions for nonlinear discontinuous strongly resonant elliptic problems

We consider quasilinear strongly resonant problems with discontinuous right-hand side. To develop an existence theory we pass to a multivalued problem by, roughly speaking, filling in the gaps at the discontinuity points. We prove the existence of at least three nontrivial solutions. Our approach uses the nonsmooth critical point theory for locally Lipschitz functionals due to Chang (1981) and a generalized version of the Ekeland variational principle. At the end of the paper we show that the nonsmooth Palais-Smale (PS)-condition implies the coercivity of the functional, extending this way a well-known result of the “smooth” case

DNA-Methylation Profiling of Fetal Tissues Reveals Marked Epigenetic Differences between Chorionic and Amniotic Samples

Epigenetic mechanisms including DNA methylation are supposed to play a key role in fetal development. Here we have investigated fetal DNA-methylation levels of 27,578 CpG loci in 47 chorionic villi (CVS) and 16 amniotic cell (AC) samples. Methylation levels differed significantly between karyotypically normal AC and CVS for 2,014 genes. AC showed more extreme DNA-methylation levels of these genes than CVS and the differentially methylated genes are significantly enriched for processes characteristic for the different cell types sampled. Furthermore, we identified 404 genes differentially methylated in CVS with trisomy 21. These genes were significantly enriched for high CG dinucleotid (CpG) content and developmental processes associated with Down syndrome. Our study points to major tissue-specific differences of fetal DNA-methylation and gives rise to the hypothesis that part of the Down syndrome phenotype is epigenetically programmed in the first trimester of pregnancy