Existence for (p, q) critical systems in the Heisenberg group

Abstract This paper deals with the existence of entire nontrivial solutions for critical quasilinear systems () in the Heisenberg group ℍn, driven by general (p, q) elliptic operators of Marcellini types. The study of () requires relevant topics of nonlinear functional analysis because of the lack of compactness. The key step in the existence proof is the concentration–compactness principle of Lions, here proved for the first time in the vectorial Heisenberg context. Finally, the constructed solution has both components nontrivial and the results extend to the Heisenberg group previous theorems on quasilinear (p, q) systems

(p,Q) systems with critical singular exponential nonlinearities in the Heisenberg group

AbstractThe paper deals with the existence of solutions for(p,Q)(p,Q)coupled elliptic systems in the Heisenberg group, with critical exponential growth at infinity and singular behavior at the origin. We derive existence of nonnegative solutions with both components nontrivial and different, that is solving an actual system, which does not reduce into an equation. The main features and novelties of the paper are the presence of a general coupled critical exponential term of the Trudinger-Moser type and the fact that the system is set inℍn{{\mathbb{H}}}^{n}

Struwe's Global Compactness and energy approximation of the critical Sobolev embedding in the Heisenberg group

We investigate some of the effects of the lack of compactness in the critical Folland-Stein-Sobolev embedding in very general (possible non-smooth) domains, by proving via De Giorgi's $\Gamma$-convergence techniques that optimal functions for a natural subcritical approximations of the Sobolev quotient concentrate energy at one point. In the second part of the paper, we try to restore the compactness by extending the celebrated Global Compactness result to the Heisenberg group via a completely different approach with respect to the original one by Struwe (Math. Z. 1984)

Concentration compactness results for systems in the Heisenberg group

In this paper we complete the study started in [P. Pucci, L. Temperini, Existence for (p,q) critical systems in the Heisenberg group, Adv. Nonlinear Anal. 9 (2020), 895-922] on some variants of the concentration-compactness principle in bounded PS domains Ω of the Heisenberg group [formula]. The concentration-compactness principle is a basic tool for treating nonlinear problems with lack of compactness. The results proved here can be exploited when dealing with some kind of elliptic systems involving critical nonlinearities and Hardy terms

Existence for (p, q) critical systems in the Heisenberg group

Abstract This paper deals with the existence of entire nontrivial solutions for critical quasilinear systems () in the Heisenberg group ℍn, driven by general (p, q) elliptic operators of Marcellini types. The study of () requires relevant topics of nonlinear functional analysis because of the lack of compactness. The key step in the existence proof is the concentration–compactness principle of Lions, here proved for the first time in the vectorial Heisenberg context. Finally, the constructed solution has both components nontrivial and the results extend to the Heisenberg group previous theorems on quasilinear (p, q) systems

Entire solutions for some critical equations in the Heisenberg group

We complete the study started in the paper [P. Pucci, L. Temperini, On the concentration-compactness principle for Folland-Stein spaces and for fractional horizontal Sobolev spaces, Math. Eng. 5 (2023), Paper no. 007], giving some applications of its abstract results to get existence of solutions of certain critical equations in the entire Heinseberg group. In particular, different conditions for existence are given for critical horizontal p-Laplacian equations

A thiabendazole sulfonamide shows potent inhibitory activity against mammalian and nematode α-carbonic anhydrases

A sulfonamide derivative of the antihelmintic drug thiabendazole was prepared and investigated for inhibition of the zinc enzyme carbonic anhydrase CA (EC 4.2.1.1). Mammalian isoforms CA I-XIV and the nematode enzyme of Caenorhabditis elegans CAH-4b were included in this study. Thiabendazole-5-sulfonamide was a very effective inhibitor of CAH-4b and CA IX (K(I)s of 6.4-9.5nm) and also inhibited effectively isozymes CA I, II, IV-VII, and XII, with K(I)s in the range of 17.8-73.2nM. The high resolution X-ray crystal structure of its adduct with isozyme II evidenced the structural elements responsible for this potent inhibitory activity

Association of maternal hypertension and chorioamnionitis with preterm outcomes

OBJECTIVES: We compared the relative effect of hypertensive disorders of pregnancy and chorioamnionitis on adverse neonatal outcomes in very preterm neonates, and studied whether gestational age (GA) modulates these effects. METHODS: A cohort of neonates 23 to 30 weeks' GA, born in 2008 to 2011 in 82 hospitals adhering to the Italian Neonatal Network, was analyzed. Infants born from mothers who had hypertensive disorders (N = 2096) were compared with those born after chorioamnionitis (N = 1510). Statistical analysis employed logistic models, adjusting for GA, hospital, and potential confounders. RESULTS: Overall mortality was higher after hypertension than after chorioamnionitis (odds ratio [OR], 1.39; 95% confidence interval [CI], 1.08-1.80), but this relationship changed across GA weeks; the OR for hypertension was highest at low GA, whereas from 28 weeks' GA onward, mortality was higher for chorioamnionitis. For other outcomes, the relative risks were constant across GA; infants born after hypertension had an increased risk for bronchopulmonary dysplasia (OR, 2.20; 95% CI, 1.68-2.88) and severe retinopathy of prematurity (OR, 1.48; 95% CI, 1.02-2.15), whereas there was a lower risk for early-onset sepsis (OR, 0.25; 95% CI, 0.19-0.34), severe intraventricular hemorrhage (OR, 0.65; 95% CI, 0.48-0.88), periventricular leukomalacia (OR, 0.70; 95% CI, 0.48-1.01), and surgical necrotizing enterocolitis or gastrointestinal perforation (OR, 0.47; 95% CI, 0.31-0.72). CONCLUSIONS: Mortality and other adverse outcomes in very preterm infants depend on antecedents of preterm birth. Hypertension and chorioamnionitis are associated with different patterns of outcomes; for mortality, the effect changes across GA weeks. Copyright \uc2\ua9 2014 by the American Academy of Pediatrics