Related papers: Large singular solutions for conformal $Q$-curvatu…

We classify entire positive singular solutions to a family of critical sixth order equations in the punctured space with a non-removable singularity at the origin. More precisely, we show that when the origin is a non-removable singularity,…

In this paper we consider the existence of positive solutions for a singular elliptic problem involving an asymtotically linear nonlinearity and depending on one positive parameter. Using variational methods, together with comparison…

Let $C_1,C_2\subseteq\mathbb{G}_m^N(\mathbb{C})$ be irreducible closed algebraic curves, with $N\geq 3$. Suppose $C_1$ is not contained in an algebraic subgroup of $\mathbb{G}_m^N(\mathbb{C})$ of dimension $1$ and $C_1\cup C_2$ is not…

Given $(M,g_0)$ a closed Riemannian manifold and a nonempty closed subset $X$ in $M$, the singular $\sigma_k-$Yamabe problem asks for a complete metric $g$ on $M\backslash X$ conformal to $g_0$ with constant $\sigma_k-$curvature. The…

It is shown that given a metric space $X$ and a $\sigma$-finite positive regular Borel measure $\mu$ on $X$, there exists a bounded continuous real-valued function on $X$ that is one-to-one on the complement of a set of $\mu$ measure zero.

Is a sequence of Riemannian manifolds with positive scalar curvature, satisfying some conditions to keep the sequence reasonable, compact? What topology should one use for the convergence and what is the regularity of the limit space? In…

This paper deals with the moment problem on a (not necessarily finitely generated) commutative unital real algebra $A$. We define moment functionals on $A$ as linear functionals which can be written as integrals over characters of $A$ with…

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…

We study the boundary behaviour of the solutions of (E) $-\Delta_p u+|\nabla u|^q=0$ in a domain $\Omega \subset \mathbb{R}^N$, when $N\geq p > q >p-1$. We show the existence of a critical exponent $q_* < p$ such that if $p-1 < q < q_*$…

Let $(M,g)$ be an analytic Riemannian manifold of dimension $n \geq 5$. In this paper, we consider the so-called constant $Q$-curvature equation \[ \varepsilon^4\Delta_{g}^2 u -\varepsilon^2 b \Delta_{g} u +a u = u^{p} , \qquad \text{in }…

We explore spherically symmetric stationary solutions, generated by ``stars'' with regular interiors, in purely massive gravity. We reexamine the claim that the resummation of non-linear effects can cure, in a domain near the source, the…

We prove two theorems about differentiable functions on the Banach space C(K), where K is compact. (i) If C(K) admits a non-trivial function of class C^m and of bounded support, then all continuous real-valued functions on C(K) may be…

We completely classify Friedmann-Lema\^{i}tre-Robertson-Walker solutions with spatial curvature $K=0,\pm 1$ and equation of state $p=w\rho$, according to their conformal structure, singularities and trapping horizons. We do not assume any…

We exhibit a class of "relatively curved" $\vec{\gamma}(t) := (\gamma_1(t),\dots,\gamma_n(t))$, so that the pertaining multi-linear maximal function satisfies the sharp range of H\"{o}lder exponents, \[ \left\| \sup_{r > 0} \ \frac{1}{r}…

In this paper, we are concerned with the following elliptic equation \begin{equation*} \begin{cases} -\Delta u= Q(x)u^{2^*-1 }+\varepsilon u^{s},~ &{\text{in}~\Omega},\\[1mm] u>0,~ &{\text{in}~\Omega},\\[1mm] u=0, &{\text{on}~\partial…

In this paper, the notion of $\mathbb{C}$-simulation function is introduced and the existence and uniqueness of common fixed points of two self-mappings satisfying contractive conditions in the setting of complex valued metric spaces via…

In this paper we establish the existence of two positive solutions for the obstacle problem $$ \displaystyle \int_{\Re}\left[u'(v-u)'+(1+\lambda V(x))u(v-u)\right] \geq \displaystyle \int_{\Re} f(u)(v-u), \forall v\in \Ka $$ where $f$ is a…

We generalise and sharpen several recent results in the literature regarding the existence and complete classification of the isolated singularities for a broad class of nonlinear elliptic equations of the form \begin{equation} -{\rm…

We study existence and nonexistence of strictly positive solutions for the elliptic problems of the form $Lu=m\left( x\right) u^{p}$ in a bounded open interval, with zero boundary conditions, where $L$ is a strongly uniformly elliptic…

We show that for a very general class of curvature functions defined in the positive cone, the problem of finding a complete strictly locally convex hypersurface in $H^n+1$ satisfying $f(\kappa)=\sigma\in(0, 1)$ with a prescribed asymptotic…