Quantization for an elliptic equation of order 2m with critical exponential non-linearity

On a smoothly bounded domain $\Omega\subset\R{2m}$ we consider a sequence of positive solutions $u_k\stackrel{w}{\rightharpoondown} 0$ in $H^m(\Omega)$ to the equation $(-\Delta)^m u_k=\lambda_k u_k e^{mu_k^2}$ subject to Dirichlet boundary conditions, where $0<\lambda_k\to 0$. Assuming that $$\Lambda:=\lim_{k\to\infty}\int_\Omega u_k(-\Delta)^m u_k dx<\infty,$$ we prove that $\Lambda$ is an integer multiple of \Lambda_1:=(2m-1)!\vol(S^{2m}), the total $Q$-curvature of the standard $2m$-dimensional sphere.Comment: 33 page

Existence of solutions to a higher dimensional mean-field equation on manifolds

For $m\geq 1$ we prove an existence result for the equation $$(-\Delta_g)^m u+\lambda=\lambda\frac{e^{2mu}}{\int_M e^{2mu}d\mu_g}$$ on a closed Riemannian manifold $(M,g)$ of dimension $2m$ for certain values of $\lambda$.Comment: 15 Page

The role of clinically significant antiphospholipid antibodies in systemic lupus erythematosus

The objective is to investigate the role of clinically significant antiphospholipid antibodies (aPL) in a cohort of systemic lupus erythematosus (SLE) patients. All SLE patients followed for at least 5 years and with available aPL profile at the beginning of the follow-up in our center were studied. Clinically significant aPL were defined as: positive lupus anticoagulant test, anti-cardiolipin and/or anti- β2Glycoprotein I IgG/IgM &gt;99th percentile on two or more occasions at least 12 weeks apart. Patients with and without clinically significant aPL were compared by univariate (Chi square or Fisher's exact test for categorical variables and Student's t or Mann-Whitney test for continuous variables) and multivariate analysis (logistic regression analysis). P values &lt;0.05 were considered significant. Among 317 SLE patients studied, 117 (37%) had a clinically significant aPL profile at baseline. Such patients showed at univariate analysis an increased prevalence of deep venous thrombosis, pulmonary embolism, cardiac valvular disease, cognitive dysfunction and antiphospholipid syndrome (APS), but a reduced prevalence of acute cutaneous lupus and anti-extractable nuclear antigens (ENA) when compared with patients without clinically significant aPL. Multivariate analysis confirmed the association between clinically significant aPL and reduced risk of acute cutaneous lupus [p=0.003, odds ratio (OR) 0.43] and ENA positivity (p&lt;0.001, OR 0.37), with increased risk of cardiac valvular disease (p=0.024, OR 3.1) and APS (p&lt;0.0001, OR 51.12). Triple positivity was the most frequent profile and was significantly associated to APS (p&lt;0.0001, OR 28.43). Our study showed that one third of SLE patients had clinically significant aPL, and that this is associated with an increased risk, especially for triple positive, of APS, and to a different clinical and serological pattern of disease even in the absence of APS

Asymptotics and quantization for a mean-field equation of higher order

Given a regular bounded domain $\Omega\subset\R{2m}$, we describe the limiting behavior of sequences of solutions to the mean field equation of order $2m$, $m\geq 1$, $$(-\Delta)^m u=\rho \frac{e^{2mu}}{\int_\Omega e^{2mu}dx}\quad\text{in}\Omega,$$ under the Dirichlet boundary condition and the bound $0<\rho\leq C$. We emphasize the connection with the problem of prescribing the $Q$-curvature.Comment: 21 page

A threshold phenomenon for embeddings of H0mH^m_0 into Orlicz spaces

We consider a sequence of positive smooth critical points of the Adams-Moser-Trudinger embedding of $H^m_0$ into Orlicz spaces. We study its concentration-compactness behavior and show that if the sequence is not precompact, then the liminf of the $H^m_0$-norms of the functions is greater than or equal to a positive geometric constant.Comment: 14 Page