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

New SX Phe variables in the globular cluster NGC 288

We report the discovery of two new variable stars in the metal-poor globular cluster NGC 288, found by means of time-series CCD photometry. We classified the new variables as SX Phoenicis due to their characteristic fundamental mode periods (1.02 +- 0.01 and 0.69 +- 0.01 hours), and refine the period estimates for other six known variables. SX Phe stars are known to follow a well-defined Period-Luminosity (P-L) relation and, thus, can be used for determining distances; they are more numerous than RR Lyraes in NGC~288. We obtain the P-L relation for the fundamental mode M_V = (-2.59 +- 0.18) log P_0(d) + (-0.34 +- 0.24) and for the first-overtone mode M_V = (-2.59 +- 0.18) log P_1(d) + (0.50 +- 0.25). Multi-chromatic isochrone fits to our UBV color-magnitude diagrams, based on the Dartmouth Stellar Evolution Database, provide = -1.3 +- 0.1, E(B-V) = 0.02 +- 0.01 and absolute distance modulus (m-M)0 = 14.72 +- 0.01 for NGC 288.Comment: 8 pages, 9 figures, 3 table

Conformal metrics on R-2m with constant Q-curvature Luca Martinazzi

We study the conformal metrics on R-2m with constant Q-curvature Q is an element of R having finite volume, particularly in the case Q &lt;= 0. We show that when Q &lt; 0 such metrics exist in R-2m if and only if m &gt; 1. Moreover, we study their asymptotic behavior at infinity, in analogy with the case Q &gt; 0, which we treated in a recent paper. When Q D 0, we show that such metrics have the form e(2p) g(R2m), where p is a polynomial such that 2 &lt;= deg p &lt;= 2 m 2 and sup(R2m) p &lt; infinity. In dimension 4, such metrics correspond to the polynomials p of degree 2 with lim(|x|-&gt;infinity) p(x) = -infinity

Extremals for Fractional Moser-Trudinger Inequalities in Dimension 1 via Harmonic Extensions and Commutator Estimates

We prove the existence of extremals for fractional Moser-Trudinger inequalities in an interval and on thewhole real line. In both caseswe use blow-up analysis for the corresponding Euler-Lagrange equation, which requires new sharp estimates obtained via commutator techniques

Normal conformal metrics on R4 with Q-curvature having power-like growth

Answering a question by M. Struwe [26] related to the blow-up behavior in the Nirenberg problem, we show that the prescribed Q-curvature equation Δ2u=(1−|x|p)e4u in R4,Λ:=∫R4(1−|x|p)e4udx<∞ has normal solutions (namely solutions which can be written in integral form, and hence satisfy Δu(x)=O(|x|−2) as |x|→∞) if and only if p∈(0,4) and [Formula presented] We also prove existence and non-existence results for the positive curvature case, namely for Δ2u=(1+|x|p)e4u in R4, and discuss some open questions