Related papers: On singular Q-curvature type equations

This paper is devoted to the construction of weak solutions to the singular constant $Q$-curvature problem. We build on several tools developed in the last years. This is the first construction of singular metrics on closed manifolds of…

We study uniqueness of positive solutions to the conformal scalar curvature equation on complete Riemannian manifolds with constant negative scalar curvature. We apply the results to show that conformal transformations on certain complete…

For dimensions $n \geq 3$, we classify singular solutions to the generalized Liouville equation $(-\Delta)^{n/2} u = e^{nu}$ on $\mathbb{R}^n \setminus \{0\}$ with the finite integral condition $\int_{\mathbb{R}^n} e^{nu} < \infty$ in terms…

We study solutions to conformally invariant equations with isolated singularties.

We consider the problem of varying conformally the metric of a four dimensional manifold in order to obtain constant $Q$-curvature. The problem is variational, and solutions are in general found as critical points of saddle type. We show…

In this article, we study higher-order Bol's inequality for radial normal solutions to a singular Liouville equation. By applying these inequalities along with compactness arguments, we derive necessary and sufficient conditions for the…

This article is dedicated to solving the Einstein constraint equations with apparent horizon boundaries and freely specified mean curvature. The main novelty is that we study the conformal constraint equations assuming only low regularity.

We define a generalization of convex functions, which we call $\delta$-convex functions, and show they must satisfy interior H\"older and $W^{1,p}$ estimates. As an application, we consider solutions of a certain class of fully nonlinear…

We consider the constant Q-curvature metric problem in the given conformal class on conic 4-manifolds and study related differential equations.

In this paper we study the existence of positive smooth solutions for a class of singular (p(x),q(x))- Laplacian systems by using sub and supersolution methods.

In this paper we classify the isolated singularities of positive solutions to Choquard equation and prove the existence of isolated singular solutions.

The paper concerns singular solutions of nonlinear elliptic equations.

A Q-homology plane is a normal complex algebraic surface having trivial rational homology. We obtain a structure theorem for Q-homology planes with smooth locus of non-general type. We show that if a Q-homology plane contains a non-quotient…

We study singularities of surfaces which are given by Kenmotsu-type formula with prescribed unbounded mean curvature.

Given a conformal metric with finite total Q-curvature, we show that the assumptions on scalar curvature sensitively govern the Q-curvature integral. Additionally, we introduce a conformal mass for such manifolds. Using such mass, we…

We consider the uniqueness of solutions of ordinary differential equations where the coefficients may have singularities. We derive upper bounds on the the order of singularities of the coefficients and provide examples to illustrate the…

In this paper, we study the isolated singularities of the conformal Gaussian curvature equation \[ -\Delta u = K(x) e^{u} \quad ~ in ~ B_{1} \setminus \{ 0 \}, \] where $B_1 \setminus \{ 0 \} \subset \mathbb{R}^2$ is the punctured unit…

In this paper, we study the existence of positive functions $K \in C^1(\mathbb{S}^n)$ such that the conformal $Q$-curvature equation \begin{equation}\label{001} P_m (v) =K v^{\frac{n+2m}{n-2m}}~~~~~~ {on} ~ \mathbb{S}^n \{equation} has a…

In this paper, using the similarity method, we construct particular solutions with singularities for degenerate high-order equations. The considered equations have singularities of the first and second kind. Particular solutions are…

We give an introductory account of functional determinants of elliptic operators on manifolds and Polyakov-type formulas for their infinitesimal and finite conformal variations. We relate this to extremal problems and to the Q-curvature on…