Abstract.Replacing linear diffusion by a degenerate diffusion of porous medium type is known to regularize the classical two-dimensional parabolic-elliptic Keller-Segel model [V. Calvez and J. A. Carrillo, J. Math. Pures Appl. (9), 86 (2006), pp. 155-175]. The implications of nonlinear diffusion are that solutions exist globally and are uniformly bounded in time. We analyze the stationary case showing the existence of a unique, up to translation, global minimizer of the associated free energy. Furthermore, we prove that this global minimizer is a radially decreasing compactly supported continuous density function which is smooth inside its support, and it is characterized as the unique compactly supported stationary state of the evolution model. This unique profile is the clear candidate to describe the long time asymptotics of the diffusion dominated classical Keller-Segel model for general initial data. 1. Introduction. Ground states of free energies play a crucial role in the long time asymptotics of nonlinear aggregation diffusion models. These nonlocal partial differential equations (PDEs) are ubiquitous in the mathematical modeling of phenomena which involve a large number of particles. For instance, nonlocal drift-diffusion equations show up naturally in semiconductor modeling, bacterial chemotaxis, granular media, and many other areas; see [18,9,13] and the references therein. These equations are just based on two competing mechanisms, namely the attraction, modeled by a nonlocal force, and the repulsion, modeled by a nonlinear diffusion.One of the archetypal models of nonlinear aggregation diffusion is the so-called classical parabolic-elliptic Keller-Segel model. This model was classically introduced as the simplest description for chemotactic bacteria movement in which the tendency of linear diffusion to spread repels the attraction due to the logarithmic kernel interaction in two dimensions. Although there is a large amount of literature in this field, many advances have been made in the last 10 years thanks to the combination of different ideas ranging from functional inequalities to entropy-entropy dissipation techniques passing through optimal transport. We refer the reader to [9, 8, 6, 11] and

In this paper we prove a Sobolev and a Morrey type inequality involving the mean curvature and the tangential gradient with respect to the level sets of the function that appears in the inequalities. Then, as an application, we establish a priori estimates for semi-stable solutions of −∆ p u = g(u) in a smooth bounded domain Ω ⊂ R n . In particular, we obtain new L r and W 1,r bounds for the extremal solution u ⋆ when the domain is strictly convex. More precisely, we prove that u ⋆ ∈ L ∞ (Ω) if n ≤ p + 2 and u ⋆ ∈ L np n−p−2 (Ω) ∩ W 1,p 0 (Ω) if n > p + 2.

MSC:primary 35J60 secondary 35B05, 35A15We prove nondegeneracy of extremals for some Hardy-SobolevMaz'ya inequalities and present applications to scalar curvaturetype problems, including the Webster scalar curvature equation in a cylindrically symmetric setting. The main theme is hyperbolic symmetry.

We obtain a generalized version of an inequality, first derived by C. Bandle in the analytic setting, for weak subsolutions of a singular Liouville-type equation. As an application we obtain a new proof of the Alexandrov isoperimetric inequality on singular abstract surfaces. Interestingly enough, motivated by this geometric problem, we obtain a seemingly new characterization of local metrics on Alexandrov's surfaces of bounded curvature. At least to our knowledge, the characterization of the equality case in the isoperimetric inequality in such a weak framework is new as well.

Abstract. Motivated by the study of self gravitating cosmic strings, we pursue the well known method by C. Bandle to obtain a weak version of the classical Alexandrov's isoperimetric inequality. In fact we derive some quantitative estimates for weak subsolutions of a Liouville-type equation with conical singularities. Actually we succeed in generalizing previously known results, including Bol's inequality and pointwise estimates, to the case where the solutions solve the equation just in the sense of distributions. Next, we derive some new pointwise estimates suitable to be applied to a class of singular cosmic string equations. Finally, interestingly enough, we apply these results to establish a minimal mass property for solutions of the cosmic string equation which are supersolutions of the singular Liouville-type equation.