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

Determinants of enhanced vulnerability to coronavirus disease 2019 in UK patients with cancer: a European study

Despite high contagiousness and rapid spread, severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) has led to heterogeneous outcomes across affected nations. Within Europe (EU), the United Kingdom (UK) is the most severely affected country, with a death toll in excess of 100,000 as of January 2021. We aimed to compare the national impact of coronavirus disease 2019 (COVID-19) on the risk of death in UK patients with cancer versus those in continental EU. Methods: We performed a retrospective analysis of the OnCovid study database, a European registry of patients with cancer consecutively diagnosed with COVID-19 in 27 centres from 27th February to 10th September 2020. We analysed case fatality rates and risk of death at 30 days and 6 months stratified by region of origin (UK versus EU). We compared patient characteristics at baseline including oncological and COVID-19-specific therapy across UK and EU cohorts and evaluated the association of these factors with the risk of adverse outcomes in multivariable Cox regression models. Findings: Compared with EU (n = 924), UK patients (n = 468) were characterised by higher case fatality rates (40.38% versus 26.5%, p < 0.0001) and higher risk of death at 30 days (hazard ratio [HR], 1.64 [95% confidence interval {CI}, 1.36-1.99]) and 6 months after COVID-19 diagnosis (47.64% versus 33.33%; p < 0.0001; HR, 1.59 [95% CI, 1.33-1.88]). UK patients were more often men, were of older age and have more comorbidities than EU counterparts (p < 0.01). Receipt of anticancer therapy was lower in UK than in EU patients (p < 0.001). Despite equal proportions of complicated COVID-19, rates of intensive care admission and use of mechanical ventilation, UK patients with cancer were less likely to receive anti-COVID-19 therapies including corticosteroids, antivirals and interleukin-6 antagonists (p < 0.0001). Multivariable analyses adjusted for imbalanced prognostic factors confirmed the UK cohort to be characterised by worse risk of death at 30 days and 6 months, independent of the patient's age, gender, tumour stage and status; number of comorbidities; COVID-19 severity and receipt of anticancer and anti-COVID-19 therapy. Rates of permanent cessation of anticancer therapy after COVID-19 were similar in the UK and EU cohorts. Interpretation: UK patients with cancer have been more severely impacted by the unfolding of the COVID-19 pandemic despite societal risk mitigation factors and rapid deferral of anticancer therapy. The increased frailty of UK patients with cancer highlights high-risk groups that should be prioritised for anti-SARS-CoV-2 vaccination. Continued evaluation of long-term outcomes is warranted

High regularity of the solution to the singular elliptic p(⋅)-Laplacian system

We study the regularity properties of solutions to the non-homogeneous singular p(x)-Laplacian system in a bounded domain of R^n. Under suitable restrictions on the range of p(x), we construct a W^{2,r} solution, with r&gt;n, that implies the Hölder continuity of the gradient. Moreover, assuming just p(x)in(1,2) we prove that the second derivatives belong to L^2

Existence and Regularity of Steady Flows for Shear-Thinning Liquids in Exterior Two-Dimensional

We show that the two-dimensional exterior boundary-value problem (flow past a cylinder) associated with a class of shear-thinning liquid models possesses at least one solution for data of arbitrary "size". This result must be contrasted with its counterpart for the Navier-Stokes model, where a similar result is known to hold, to date, only if the size of the data is sufficiently restricted

On the C^{1,\gamma}(\overline\Omega)\cap W^{2,2}(\Omega) regularity for a class of electro-rheological fluids

We are concerned with a system of nonlinear partial differential equations with p(x)-structure, 1 < p(infinity) <= p(x) <= p(0) < +infinity, and no-slip boundary conditions. We prove the existence and uniqueness of a C-1,C-gamma ((Omega) over bar) boolean AND W-2,W-2 (Omega) solution corresponding to small data, without further restrictions on the bounds p(infinity), p(0). In particular this result is applicable to the steady motion of shear-dependent electro-rheological fluids