Four solutions of an inhomogeneous elliptic equation with critical exponent and singular term

In this paper, we prove the existence of four nontrivial solutions of a elliptic equation with a Hardy singularity and show that at least one of them is sign changing. Our results extend some previous works on the literature, as Tarantello(1993), Kang-Deng(2005) and Hirano-Shioji(2005)

An improved geometric inequality via vanishing moments, with applications to singular Liouville equations

We consider a class of singular Liouville equations on compact surfaces motivated by the study of Electroweak and Self-Dual Chern-Simons theories, the Gaussian curvature prescription with conical singularities and Onsager's description of turbulence. We analyse the problem of existence variationally, and show how the angular distribution of the conformal volume near the singularities may lead to improvements in the Moser-Trudinger inequality, and in turn to lower bounds on the Euler-Lagrange functional. We then discuss existence and non-existence results.Comment: some references adde

New improved Moser-Trudinger inequalities and singular Liouville equations on compact surfaces

We consider a singular Liouville equation on a compact surface, arising from the study of Chern-Simons vortices in a self dual regime. Using new improved versions of the Moser-Trudinger inequalities (whose main feature is to be scaling invariant) and a variational scheme, we prove new existence results.Comment: to appear in GAF

TGF-β concentrations and activity are down-regulated in the aqueous humor of patients with neovascular age-related macular degeneration

Controversy still exists regarding the role of the TGF-β in neovascular age-related macular degeneration (nAMD), a major cause of severe visual loss in the elderly in developed countries. Here, we measured the concentrations of active TGF-β1, TGF-β2, and TGF-β3 by ELISA in the aqueous humor of 20 patients affected by nAMD, who received 3 consecutive monthly intravitreal injections of anti-VEGF-A antibody. Samples were collected at baseline (before the first injection), month 1 (before the second injection), and month 2 (before the third injection). The same samples were used in a luciferase-based reporter assay to test the TGF-β pathway activation. Active TGF-β1 concentrations in the aqueous humor were below the minimum detectable dose. Active TGF-β2 concentrations were significantly lower at baseline and at month 1, compared to controls. No significant differences in active TGF-β3 concentration were found among the sample groups. Moreover, TGF-β pathway activation was significantly lower at baseline compared to controls. Our data corroborate an anti-angiogenic role for TGF-β2 in nAMD. This should be considered from the perspective of a therapy using TGF-β inhibitors

A guide to the Choquard equation

We survey old and recent results dealing with the existence and properties of solutions to the Choquard type equations $$-\Delta u + V(x)u = \bigl(|x|^{-(N-\alpha)} * |u|^p\bigr)|u|^{p - 2} u \qquad \text{in $\mathbb{R}^N$},$$ and some of its variants and extensions.Comment: 39 page