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We consider a class of semilinear equations with an absorption nonlinear zero order term of power type, where elliptic condition is given in terms of Gauss measure. In the case of the superlinear equation we introduce a suitable definition of solutions in order to prove the existence and uniqueness of a solution in ℝN without growth restrictions at infinity. A comparison result in terms of the half-space Gaussian symmetrized problem is also proved. As an application, we give some estimates in measure of the growth of the solution near the boundary of its support for sublinear equations. Finally we generalize our results to problems with a nonlinear zero order term not necessary of power type.

Maddalena Frau,1 Claudio F Feo,2 Francesco Feo,1 Rosa M Pascale11Department of Clinical and Experimental Medicine, Division of Experimental Pathology and Oncology, 2Department of Clinical and Experimental Medicine, Division of Surgery. University of Sassari, Sassari, ItalyAbstract: Emerging evidence assigns to epigenetic mechanisms heritable differences in gene function that come into being during cell development or via the effect of environmental factors. Epigenetic deregulation is strongly involved in the development of hepatocellular carcinoma (HCC). It includes changes in methionine metabolism, promoter hypermethylation, or increased proteasomal degradation of oncosuppressors, as well as posttranscriptional deregulation by microRNA or messenger RNA (mRNA) binding proteins. Alterations in the methylation of the promoter of methyl adenosyltransferase MAT1A and MAT2A genes in HCC result in decreased S-adenosylmethionine level, global DNA hypomethylation, and deregulation of signal transduction pathways linked to methionine metabolism and methyl adenosyltransferases activity. Changes in S-adenosylmethionine levels may also depend on MAT1A mRNA destabilization associated with MAT2A mRNA stabilization by specific proteins. Decrease in MAT1A expression has also been attributed to miRNA upregulation in HCC. A complex deregulation of miRNAs is also strongly involved in hepatocarcinogenesis, with up-regulation of different miRNAs targeting oncosuppressor genes and down-regulation of miRNAs targeting genes involved in cell-cycle and signal transduction control. Oncosuppressor gene down-regulation in HCC is also induced by promoter hypermethylation or posttranslational deregulation, leading to proteasomal degradation. The role of epigenetic changes in hepatocarcinogenesis has recently suggested new promising therapeutic approaches for HCC on the basis of the administration of methylating agents, inhibition of methyl adenosyltransferases, and restoration of the expression of tumor-suppressor miRNAs.Keywords: hepatocarcinogenesis, DNA methylation, microRNA, Piwi-interacting RNAs, stem cells, therapeutic targets