We consider the following coupled Schr$ \ddot{o} $dinger system with critical exponent in $ \mathbb{R}^3: $$ \left \{ \begin{aligned} &-\Delta u+\lambda V(|y|)u = K_1(|y|)u^5+u^2v^3,\qquad &\text{ } \mathbb{R}^3\backslash B_\epsilon(0),\\ v+\lambda V(|y|)v K_2(|y|)v^5+v^2u^3, \qquad &u >0, v>0, \quad B_\epsilon(0), \\ & (u,v) (0,0), &\text{on \partial B_\epsilon (0), &...

We prove some old and new isoperimetric inequalities with the best constant using the ABP method applied to an appropriate linear Neumann problem. More precisely, we obtain a new family of sharp isoperimetric inequalities with weights (also called densities) in open convex cones of R. Our result applies to all nonnegative homogeneous weights satisfying a concavity condition in the cone. Remarka...

In this paper we discuss four early F-type variable stars whose periods are an order of magnitude slower than known pulsators of comparable luminosity. They cannot be stars undergoing simple radial pulsations. For one or more of these stars we can discount the possibility that the variability is due to rotational modulation of star spots, interactions with (or tidal distortions by) a close comp...