Existence of solutions for a perturbed Dirichlet problem without growth conditions

AbstractWe present some results on the existence and multiplicity of solutions for boundary value problems involving equations of the type −Δu=f(x,u)+λg(x,u), where Δ is the Laplacian operator, λ is a real parameter and f,g:Ω×R→R, Ω⊂RN are two Carathéodory functions having no growth conditions with respect to the second variable. The approach is variational and mainly based on a critical point theorem by B. Ricceri

On a Perturbed Dirichlet Problem for a Nonlocal Differential Equation of Kirchhoff Type

Abstract We study the existence of positive solutions to the following nonlocal boundary value problem in , on , where , is a Carathéodory function, is a positive continuous function, and is a real parameter. Direct variational methods are used. In particular, the proof of the main result is based on a property of the infimum on certain spheres of the energy functional associated to problem in ,

Remarks on a multiplicity theorem for a perturbed Neumann problem

AbstractThe aim of this article is to prove that, given two potential functionals Ψ1, Ψ2 on W1,2(Ω) which coincide on a set of the type {u∈W1,2(Ω):a⩽u(x)⩽ba.e. inΩ}, then under suitable summability conditions, certain local minima of Ψ1 are local minima for Ψ2 as well. An application of this result allows us to obtain a multiplicity theorem for a Neumann problem where we impose a less restrictive oscillating behavior on the nonlinearity than the one required in an analogous result recently established by B. Ricceri

An existence theorem for an implicit integral equation with discontinuous right-hand side

We establish a result concerning the existence of solutions for the following implicit integral equation: , where is not supposed continuous with respect to the second variable

cis-Regulatory sequences driving the expression of the Hbox12 homeobox-containing gene in the presumptive aboral ectoderm territory of the Paracentrotus lividus sea urchin embryo.

Embryonic development is coordinated by networks of evolutionary conserved regulatory genes encoding transcription factors and components of cell signalling pathways. In the sea urchin embryo, a number of genes encoding transcription factors display territorial restricted expression. Among these, the zygotic Hbox12 homeobox gene is transiently transcribed in a limited number of cells of the animal-lateral half of the early Paracentrotus lividus embryo, whose descendants will constitute part of the ectoderm territory. To obtain insights on the regulation of Hbox12 expression, we have explored the cis-regulatory apparatus of the gene. In this paper, we show that the intergenic region of the tandem Hbox12 repeats drives GFP expression in the presumptive aboral ectoderm and that a 234 bp fragment, defined aboral ectoderm (AE) module, accounts for the restricted expression of the transgene. Within this module, a consensus sequence for a Sox factor and the binding of the Otx activator are both required for correct Hbox12 gene expression. Spatial restriction to the aboral ectoderm is achieved by a combination of different repressive sequence elements. Negative sequence elements necessary for repression in the endomesoderm map within the most upstream 60 bp region and nearby the Sox binding site. Strikingly, a Myb-like consensus is necessary for repression in the oral ectoderm, while down-regulation at the gastrula stage depends on a GA-rich region. These results suggest a role for Hbox12 in aboral ectoderm specification and represent our first attempt in the identification of the gene regulatory circuits involved in this process

Identification and characterization of PlAlix, the Alix homologue from the Mediterranean sea urchin Paracentrotus lividus.

The sea urchin provides a relatively simple and tractable system for analyzing the early stages of embryo development. Here, we use the sea urchin species, Paracentrotus lividus, to investigate the role of Alix in key stages of embryogenesis, namely the egg fertilization and the first cleavage division. Alix is a multifunctional protein involved in different cellular processes including endocytic membrane trafficking, filamentous (F)-actin remodeling, and cytokinesis. Alix homologues have been identified in different metazoans; in these organisms, Alix is involved in oogenesis and in determination/differentiation events during embryo development. Herein, we describe the identification of the sea urchin homologue of Alix, PlAlix. The deduced amino acid sequence shows that Alix is highly conserved in sea urchins. Accordingly, we detect the PlAlix protein cross-reacting with monoclonal Alix antibodies in extracts from P. lividus, at different developmental stages. Focusing on the role of PlAlix during early embryogenesis we found that PlAlix is a maternal protein that is expressed at increasingly higher levels from fertilization to the 2-cell stage embryo. In sea urchin eggs, PlAlix localizes throughout the cytoplasm with a punctuated pattern and, soon after fertilization, accumulates in larger puncta in the cytosol, and in microvilli-like protrusions. Together our data show that PlAlix is structurally conserved from sea urchin to mammals and may open new lines of inquiry into the role of Alix during the early stages of embryo development

Nonlinear problems on the Sierpi\'nski gasket

This paper concerns with a class of elliptic equations on fractal domains depending on a real parameter. Our approach is based on variational methods. More precisely, the existence of at least two non-trivial weak (strong) solutions for the treated problem is obtained exploiting a local minimum theorem for differentiable functionals defined on reflexive Banach spaces. A special case of the main result improves a classical application of the Mountain Pass Theorem in the fractal setting, given by Falconer and Hu (1999)

Insulin Secretory Function Is Impaired in Isolated Human Islets Carrying the Gly972→Arg IRS-1 Polymorphism

Type 2 (non–insulin-dependent) diabetes results from decreased insulin action in peripheral target tissues (insulin resistance) and impaired pancreatic β-cell function. These defects reflect both genetic components and environmental risk factors. Recently, the common Gly972→Arg amino acid polymorphism of insulin receptor substrate 1 (Arg972 IRS-1) has been associated with human type 2 diabetes. In this study, we report on some functional and morphological properties of isolated human islets carrying the Arg972 IRS-1 polymorphism. Insulin content was lower in variant than control islets (94 ± 47 vs. 133 ± 56 μU/islet; P < 0.05). Stepwise glucose increase (1.7 to 16.7 mmol/l) significantly potentiated insulin secretion from control islets, but not Arg972 IRS-1 islets, with the latter also showing a relatively lower response to glyburide and a significantly higher response to arginine. Proinsulin release mirrored insulin secretion, and the insulin-to-proinsulin ratio in response to arginine was significantly lower from Arg972 IRS-1 islets than from control islets. Glucose utilization and oxidation did not differ in variant and wild-type islets at both low and high glucose levels. Electron microscopy showed that Arg972 IRS-1 β-cells had a severalfold greater number of immature secretory granules and a lower number of mature granules than control β-cells. In conclusion, Arg972 IRS-1 islets have reduced insulin content, impaired insulin secretion, and a lower amount of mature secretory granules. These alterations may account for the increased predisposition to type 2 diabetes in individuals carrying the Gly972→Arg amino acid polymorphism of IRS-1

Upper gut heat shock proteins HSP70 and GRP78 promote insulin resistance, hyperglycemia, and non-alcoholic steatohepatitis

A high-fat diet increases the risk of insulin resistance, type-2 diabetes, and non-alcoholic steato-hepatitis. Here we identified two heat-shock proteins, Heat-Shock-Protein70 and Glucose-Regulated Protein78, which are increased in the jejunum of rats on a high-fat diet. We demonstrated a causal link between these proteins and hepatic and whole-body insulin-resistance, as well as the metabolic response to bariatric/metabolic surgery. Long-term continuous infusion of Heat-Shock-Protein70 and Glucose-Regulated Protein78 caused insulin-resistance, hyperglycemia, and non-alcoholic steato-hepatitis in rats on a chow diet, while in rats on a high-fat diet continuous infusion of monoclonal antibodies reversed these phenotypes, mimicking metabolic surgery. Infusion of these proteins or their antibodies was also associated with shifts in fecal microbiota composition. Serum levels of Heat-Shock-Protein70 and Glucose-Regulated Protein78were elevated in patients with non-alcoholic steato-hepatitis, but decreased following metabolic surgery. Understanding the intestinal regulation of metabolism may provide options to reverse metabolic diseases