Positive solutions for singularly perturbed nonlinear elliptic problem on manifolds via Morse theory

Given (M, g0) we consider the problem -{\epsilon}^2Delta_{g0+h}u + u = (u+)^{p-1} with ({\epsilon}, h) \in (0, {\epsilon}0) \times B{\rho}. Here B{\rho} is a ball centered at 0 with radius {\rho} in the Banach space of all Ck symmetric covariant 2-tensors on M. Using the Poincar\'e polynomial of M, we give an estimate on the number of nonconstant solutions with low energy for ({\epsilon}, h) belonging to a residual subset of (0, {\epsilon}0) \times B{\rho}, for ({\epsilon}0, {\rho}) small enough

Semiclassical limit for the nonlinear Klein Gordon equation in bounded domains

We are interested to the existence of standing waves for the nonlinear Klein Gordon equation {\epsilon}^2{\box}{\psi} + W'({\psi}) = 0 in a bounded domain D. The main result of this paper is that, under suitable growth condition on W, for {\epsilon} sufficiently small, we have at least cat(D) standing wavesfor the equation ({\dag}), while cat(D) is the Ljusternik-Schnirelmann category

Multiple solutions and profile description for a nonlinear Schr\"odinger-Bopp-Podolsky-Proca system on a manifold

We prove a multiplicity result for \begin{equation*} \begin{cases} -\varepsilon^{2}\Delta_g u+\omega u+q^{2}\phi u=|u|^{p-2}u\\[1mm] -\Delta_g \phi +a^{2}\Delta_g^{2} \phi + m^2 \phi =4\pi u^{2} \end{cases} \text{ in }M, \end{equation*} where $(M,g)$ is a smooth and compact $3$-dimensional Riemannian manifold without boundary, $p\in(4,6)$, $a,m,q\neq 0$, $\varepsilon>0$ small enough. The proof of this result relies on Lusternik-Schnirellman category. We also provide a profile description for low energy solutions.Comment: 26 page

A compactness result for scalar-flat metrics on low dimensional manifolds with umbilic boundary

Let (M,g) a compact Riemannian $n$-dimensional manifold with umbilic boundary. It is well know that, under certain hypothesis, in the conformal class of g there are scalar-flat metrics that have the boundary of M as a constant mean curvature hypersurface. In this paper we prove that these metrics are a compact set in the case of low dimensional manifolds, that is n=6,7,8, provided that the Weyl tensor is always not vanishing on the boundary.Comment: arXiv admin note: text overlap with arXiv:1903.10990, arXiv:1912.1148

Determination of sevoflurane and isopropyl alcohol in exhaled breath by thermal desorption gas chromatography-mass spectrometry for exposure assessment of hospital staff

Volatile anaesthetics and disinfection chemicals pose ubiquitous inhalation and dermal exposure risks in hospital and clinic environments. This work demonstrates specific non-invasive breath biomonitoring methodology for assessing staff exposures to sevoflurane (SEV) anaesthetic, documenting its metabolite hexafluoroisopropanol (HFIP) and measuring exposures to isopropanol (IPA) dermal disinfection fluid. Methods are based on breath sample collection in Nalophan bags, followed by an aliquot transfer to adsorption tube, and subsequent analysis by thermal desorption gas chromatography-mass spectrometry (TD-GC-MS). Ambient levels of IPA were also monitored. These methods could be generalized to other common volatile chemicals found in medical environments. Calibration curves were linear (r2=0.999) in the investigated ranges: 0.01-1000ppbv for SEV, 0.02-1700ppbv for IPA, and 0.001-0.1ppbv for HFIP. The instrumental detection limit was 10pptv for IPA and 5pptv for SEV, both estimated by extracted ion-TIC chromatograms, whereas the HFIP minimum detectable concentration was 0.5pptv as estimated in SIM acquisition mode. The methods were applied to hospital staff working in operating rooms and clinics for blood draws. SEV and HFIP were present in all subjects at concentrations in the range of 0.7-18, and 0.002-0.024ppbv for SEV and HFIP respectively. Correlation between IPA ambient air and breath concentration confirmed the inhalation pathway of exposure (r=0.95, p<0.001) and breath-borne IPA was measured as high as 1500ppbv. The methodology is easy to implement and valuable for screening exposures to common hospital chemicals. Although the overall exposures documented were generally below levels of health concern in this limited study, outliers were observed that indicate potential for acute exposures