We consider the problem of finding a metric in a given conformal class with prescribed non-positive scalar curvature and non-positive boundary mean curvature on an asymptotically Euclidean manifold with inner boundary. We obtain a necessary and sufficient condition in terms of a conformal invariant of the zero sets of the target curvatures for the existence of solutions to the problem and use this result to establish the Yamabe classification of metrics in those manifolds with respect to the solvability of the prescribed curvature problem.
Our primary purpose is to study a class of strongly coupled nonlinear elliptic systems with critical growth in a compact Riemannian manifold with constant scalar curvature. Using a gluing technique and perturbation arguments, we show the existence of singular solutions asymptotic to a Fowler-type solution near the isolated singularity.
We study positive solutions of the Yamabe equation with isolated singularity and prove the existence of solutions with prescribed asymptotic expansions near singular points and an arbitrarily high order of approximation.
We prove a necessary and sufficient condition for an asymptotically Euclidean manifold to be conformally related to one with specified nonpositive scalar curvature: the zero set of the desired scalar curvature must have a positive Yamabe invariant, as defined in the article. We show additionally how the sign of the Yamabe invariant of a measurable set can be computed from the sign of certain generalized "weighted" eigenvalues of the conformal Laplacian. Using the prescribed scalar curvature result we give a characterization of the Yamabe classes of asymptotically Euclidean manifolds. We also s…
Let $(X, g^+)$ be an asymptotically hyperbolic manifold and $(M, [\hat{h}])$ its conformal infinity. Our primary aim in this paper is to introduce the prescribed fractional scalar curvature problem on $M$ and provide solutions under various geometric conditions on $X$ and $M$. We also obtain the existence results for the fractional Yamabe problem in the endpoint case, e.g., $n = 3$, $γ= 1/2$ and $M$ is non-umbilic, etc. Every solution we find turns out to be smooth on $M$.
We study the Yamabe flow on asymptotically flat manifolds with non-positive Yamabe constant $Y\leq 0$. Previous work by the second and third named authors \cite{ChenWang} showed that while the Yamabe flow always converges in a global weighted sense when $Y>0$, the flow must diverge when $Y\leq 0$. We show here in the $Y\leq 0$ case however that after suitable rescalings, the Yamabe flow starting from any asymptotically flat manifold must converge to the unique positive function which solves the Yamabe problem on a compactification of the original manifold.
We consider, in the Euclidean setting, a conformal Yamabe-type equation related to a potential generalization of the classical constant scalar curvature problem and which naturally arises in the study of Ricci solitons structures. We prove existence and nonexistence results, focusing on the radial case, under some general hypothesis on the potential.
We prove local pointwise second derivative estimates for positive $W^{2,p}$ solutions to the $σ_k$-Yamabe equation on Euclidean domains, addressing both the positive and negative cases. Generalisations for augmented Hessian equations are also considered.
We construct solutions for the fractional Yamabe problem that are singular at a prescribed number of isolated points. This seems to be the first time that a gluing method is successfully applied to a non-local problem. The main step is an infinite-dimensional Lyapunov-Schmidt reduction method, that reduces the problem to an (infinite dimensional) Toda type system.
We give some a priori estimates of type sup*inf for Yamabe and prescribed scalar curvature type equations on Riemannian manifolds of dimension >2. The product sup*inf is caracteristic of those equations, like the usual Harnack inequalities for non negative harmonic functions. First, we have a lower bound for sup*inf for some classes of PDE on compact manifolds (like prescribed scalar cuvature). We also have an upper bound for the same product but on any Riemannian manifold not necessarily compact. An application of those result is an uniqueness solution for some PDE.
We study the Yamabe equation in the Euclidean half-space. We prove that any sign-changing solution has at least twice the energy of a standard bubble. Moreover, a sharper energy lower bound of the sign-changing solution set is also established via the method of moving planes. This bound increases the energy range for which Palais-Smale sequences of related variational problem has a non-trivial weak limit.
Given a compact Riemannian manifold $(M,g)$ without boundary of dimension $m\geq 3$ and under some symmetry assumptions, we establish existence of one positive and multiple nodal solutions to the Yamabe-type equation $$-div_{g}(a\nabla u)+bu=c|u|^{2^{\ast}-2}u\quad on\ M$$ where $a,b,c\in C^{\infty}(M)$, $a$ and $c$ are positive, $-div_{g}(a\nabla)+b$ is coercive, and $2^{\ast}=\frac{2m}{m-2}$ is the critical Sobolev exponent. In particular, if $R_{g}$ denotes the scalar curvature of $(M,g)$, we give conditions which guarantee that the Yamabe problem $$Δ_{g}u+\frac{m-2}{4(m-1} R_{g}u=κu^{2^{\a…
We describe and partially solve a natural Yamabe-type problem on smooth metric measure spaces which interpolates between the Yamabe problem and the problem of finding minimizers for Perelman's $ν$-entropy. This problem reduces in all dimensions on Euclidean space to the characterization of the minimizers of the family of Gagliardo--Nirenberg--Sobolev inequalities studied by Del Pino and Dolbeault. We show that minimizers always exist on a compact manifold provided the so-called weighted Yamabe constant is strictly less than its value on Euclidean space. We also show that strict inequality hold…
We consider the problem of prescribing the scalar and boundary mean curvatures via conformal deformation of the metric on a $n-$ dimensional compact Riemannian manifold. We deal with the case of negative scalar curvature $K$ and boundary mean curvature $H$ of arbitrary sign which are non-constant and $\mathfrak D_n=\sqrt{n(n-1)}{|K|}^{-1/2}>1$ at some point of the boundary. It is known that this problem admits a positive mountain pass solution if $n=3$, while no existence results are known for $n\geq 4$. We will consider a perturbation of the geometric problem and show the existence of a posit…
Let $G=(V,E)$ be a locally finite graph, $Ω\subset V$ be a bounded domain, $Δ$ be the usual graph Laplacian, and $λ_1(Ω)$ be the first eigenvalue of $-Δ$ with respect to Dirichlet boundary condition. Using the mountain pass theorem due to Ambrosetti-Rabinowitz, we prove that if $α<λ_1(Ω)$, then for any $p>2$, there exists a positive solution to $-Δu-αu=|u|^{p-2}u$ in $Ω^\circ$, $u=0$ on $\partialΩ$, where $Ω^\circ$ and $\partialΩ$ denote the interior and the boundary of $Ω$ respectively. Also we consider similar problems involving the $p$-Laplacian and poly-Laplacian by the same method. Such p…
We consider a closed cohomogeneity one Riemannian manifold $(M^n,g) $ of dimension $n\geq 3$. If the Ricci curvature of $M$ is positive, we prove the existence of infinite nodal solutions for equations of the form $-Δ_g u + λu = λu^q$ with $λ>0$, $q>1$. In particular for a positive Einstein manifold which is of cohomogeneity one or fibers over a cohomogeniety one Einstein manifold we prove the existence of infinite nodal solutions for the Yamabe equation, with a prescribed number of connected components of its nodal domain.
We consider the semi-Riemannian Yamabe type equations of the form \[ -\square u + λu = μ\vert u\vert^{p-1}u\quad\text{ on }M \] where $M$ is either the semi-Euclidean space or the pseudosphere of dimension $m\geq 3$, $\square$ is the semi-Riemannian Laplacian in $M$, $λ\geq0$, $μ\in\mathbb{R}\smallsetminus\{0\}$ and $p>1$. Using semi-Riemannian isoparametric functions on $M$, we reduce the PDE into a generalized Emden-Fowler ODE of the form \[ w''+q(r)w'+λw = μ\vert w\vert^{p-1}w\quad\text{ on } I, \] where $I\subset\mathbb{R}$ is $[0,\infty)$ or $[0,π]$, $q(r)$ blows-up at $0$ and $w$ is subj…
We consider a kind of Yamabe problem whose scalar curvature vanishes in the unit ball $\mathbb{B}^n$ and on the boundary $\mathbb{S}^{n-1}$ the mean curvature is prescribed. By combining critical points at infinity approach with Morse theory we obtain new existence results in higher dimensional case $n\geq 5$, under suitable pinching conditions on the mean curvature function.
This paper deals with the prescribed mean curvature equations both in the Euclidean case and in the Lorentz-Minkowski case in presence of a nonlinearity $g$ such that $g'(0)>0$. We show the existence of oscillating solutions, namely with an unbounded sequence of zeros. We show the existence of oscillating solutions, namely with an unbounded sequence of zeros. Moreover these solutions are periodic, if $N=1$, while they are radial symmetric and decay to zero at infinity with their derivatives, if $N\ge 2$.
We give sufficient and "almost" necessary conditions for the prescribed scalar curvature problems within the conformal class of a Riemannian metric $ g $ for both closed manifolds and compact manifolds with boundary, including the interesting cases $ \mathbb{S}^{n} $ or some quotient of $ \mathbb{S}^{n} $, in dimensions $ n \geqslant 3 $, provided that the first eigenvalues of conformal Laplacian (with appropriate boundary conditions if necessary) are positive. When the manifold is not some quotient of $ \mathbb{S}^{n} $, we show that, on one hand, any smooth function that is a positive consta…
We answer affirmatively a question of Aviles posed in 1983, concerning the construction of singular solutions of semilinear equations without using phase-plane analysis. Fully exploiting the semilinearity and the stability of the linearized operator in any dimension, our techniques involve a careful gluing in weighted $L^\infty$ spaces that handles multiple occurrences of criticality, without the need of derivative estimates. The above solution constitutes an \emph{Ansatz} for the Yamabe problem with a prescribed singular set of maximal dimension $(n-2)/2$, for which, using the same machinery,…
We clarify how close a second order fully nonlinear equation can come to uniform ellipticity, through counting large eigenvalues of the linearized operator. This suggests an effective and novel way to understand the structure of fully nonlinear equations of elliptic and parabolic type. As applications, we solve a fully nonlinear version of the Loewner-Nirenberg problem and a noncompact complete version of fully nonlinear Yamabe problem. Our method is delicate as shown by a topological obstruction.
From fractional Lane-Emden-Serrin equation -- existence, multiplicity and local behaviors via classical ODE -- to fractional Yamabe metrics with singularity of "maximal" dimension
In the previous paper, it has been proved that the generalized rotational hypersurfaces of O(n-1)-type and O (l+1) x O(m+1)-type, for which the mean curvature is any prescribed continuous function. This paper is a sequel, and a similar existence result is shown for any type.
This article uses the iterative schemes and perturbation methods to completely solve the Han-Li conjecture, i.e. the general boundary Yamabe problem with prescribed constant scalar curvature and constant mean curvature on compact manifolds $ (M, \partial M, g) $ with non-empty smooth boundary, $ \dim M \geqslant 3 $. It is equivalent to show the existence of a real, positive, smooth solution of $ -\frac{4(n -1)}{n - 2} Δ_{g} u + R_{g} u = λu^{\frac{n+2}{n - 2}} $ in $ M $, $ \frac{\partial u}{\partial ν} + \frac{n-2}{2} h_{g} u = \frac{n-2}{2} ζu^{\frac{n}{n - 2}} $ on $ \partial M $ with some…
Let $m\ge 2$ be an integer. For any open domain $Ω\subset\mathbb{R}^{2m}$, non-positive function $φ\in C^\infty(Ω)$ such that $Δ^m φ\equiv 0$, and bounded sequence $(V_k)\subset L^\infty(Ω)$ we prove the existence of a sequence of functions $(u_k)\subset C^{2m-1}(Ω)$ solving the Liouville equation of order $2m$ $$(-Δ)^m u_k = V_ke^{2mu_k}\quad \text{in }Ω, \quad \limsup_{k\to\infty} \int_Ωe^{2mu_k}dx<\infty,$$ and blowing up exactly on the set $S_φ:=\{x\in Ω:φ(x)=0\}$, i.e. $$\lim_{k\to\infty} u_k(x)=+\infty \text{ for }x\in S_φ \text{ and }\lim_{k\to\infty} u_k(x)=-\infty \text{ for }x\in Ω\s…
In this article, we first show that for all compact Riemannian manifolds with non-empty smooth boundary and dimension at least 3, there exists a metric, pointwise conformal to the original metric, with constant scalar curvature in the interior, and constant scalar curvature on the boundary by considering the boundary as a manifold of its own with dimension at least 2. We then show a series of prescribed scalar curvature results in the interior and on the boundary, with pointwise conformal deformation. These type of results is both an analogy and an extension of Kazdan and Warner's "Trichotomy …
By using variational techniques we provide new existence results for Yamabe-type equations with subcritical perturbations set on a compact $d$-dimensional ($d\geq 3$) Riemannian manifold without boundary. As a direct consequence of our main theorems, we prove the existence of at least one solution to the following singular Yamabe-type problem $$ \left\lbrace \begin{array}{ll} -Δ_g w + α(σ)w = μK(σ) w^\frac{d+2}{d-2} +λ\left( w^{r-1} + f(w)\right), \quad σ\in\mathcal{M} &\\ &\\ w\in H^2_α(\mathcal{M}), \quad w>0 \ \ \mbox{in} \ \ \mathcal{M} & \end{array} \right.$$ where, as usual, $Δ_g$ denote…
We consider the following prescribed boundary mean curvature problem in $\mathbb B^N$ with the Euclidean metric $-Δu =0$, $u>0$ in $B^N, \frac{\partial u}{\partialν} + \frac{N-2}{2} u =\frac{N-2}{2} K(x) u^{N/(N-2)}$ on $S^{N-1}, where $K$ is positive and rotationally symmetric on $\mathbb S^{N-1}$. We show that if ${K}$ has a local maximum point, then the equation has {\bf infinitely many positive} solutions, which are non-radial on $\mathbb S^{N-1}$.
We prove an existence theorem for convex hypersurfaces of prescribed Gauss curvature in the complement of a compact set in Euclidean space which are close to a cone.
In this paper, we are concerned with the global structure of radial solutions, with prescribed nodal properties, to the boundary value problem $$\text{div}\big(ϕ_{N}(\nabla v)\big)+λf(|x|, v)=0 ~~~\text{in} ~~B(R), ~~~ v=0 ~~~\text{on} ~~\partial B(R), $$ where $ϕ_{N}(y)=\frac{y}{\sqrt{1-|y|^{2}}},\; y\in \mathbb{R}^{N}$, $λ$ is a positive parameter, $B(R)=\{x\in \mathbb{R}^{N} :|x|<R\}$, and $|\cdot|$ denote the Euclidean norm in $\mathbb{R}^{N}$. All results, depending on the behavior of nonlinear term $f$ near 0, are obtained by using global bifurcation techniques.
In this paper, we are concerned with the global structure of radial positive solutions of boundary value problem$$\text{div}\big(ϕ_{N}(\nabla v)\big)+λf(|x|, v)=0 \text{in} B(R), v=0 \text{on} \partial B(R), $$where $ϕ_{N}(y)=\frac{y}{\sqrt{1-|y|^{2}}}, y\in \mathbb{R}^{N}$, $λ$ is a positive parameter, $B(R)=\{x\in \mathbb{R}^{N} :|x|<R\}$, and $|\cdot|$ denote the Euclidean norm in $\mathbb{R}^{N}$. All results, depending on the behavior of nonlinear term $f$ near 0, are obtained by using global bifurcation techniques.
For each set of (freely chosen) seed data, the conformal method reduces the Einstein constraint equations to a system of elliptic equations, the conformal constraint equations. We prove an admissibility criterion, based on a (conformal) prescribed scalar curvature problem, which provides a necessary condition on the seed data for the conformal constraint equations to (possibly) admit a solution. We then consider sets of asymptotically Euclidean (AE) seed data for which solutions of the conformal constraint equations exist, and examine the blowup properties of these solutions as the seed data s…
For a smooth, closed and uniformly $h$-convex hypersurface $M$ in $\mathbb{H}^{n+1}$, the horospherical Gauss map $G: M \rightarrow \mathbb{S}^n$ is a diffeomorphism. We consider the problem of finding a smooth, closed and uniformly $h$-convex hypersurface $M\subset \mathbb{H}^{n+1}$ whose $k$-th shifted mean curvature $\widetilde{H}_{k}$ ($1\leq k\leq n$) is prescribed as a positive function $\tilde{f}(x)$ defined on $\mathbb{S}^n$, i.e. \begin{eqnarray*} \widetilde{H}_{k}(G^{-1}(x))=\tilde{f}(x). \end{eqnarray*} We can prove the existence of solution to this problem if the given function $\t…
The gluing technique is used to construct hypersurfaces in Euclidean space having approximately constant prescribed mean curvature. These surfaces are perturbations of unions of finitely many spheres of the same radius assembled end-to-end along a line segment. The condition on the existence of these hypersurfaces is the vanishing of the sum of certain integral moments of the spheres with respect the prescribed mean curvature function.
We construct a sequence of compact, oriented, embedded, two-dimensional surfaces of genus one into Euclidean 3-space with prescribed, almost constant, mean curvature of the form $H(X)=1+{A}{|X|^{-γ}}$ for $|X|$ large, when $A<0$ and $γ\in(0,2)$. Such surfaces are close to sections of unduloids with small necksize, folded along circumferences centered at the origin and with larger and larger radii. The construction involves a deep study of the corresponding Jacobi operators, an application of the Lyapunov-Schmidt reduction method and some variational argument.
We study metrics of constant $Q$-curvature in the Euclidean space with a prescribed singularity at the origin, namely solutions to the equation $$(-Δ)^\frac{n}{2}w=e^{nw}-cδ_{0} \text{ on } \mathbb R^n,$$ under a finite volume condition. We analyze the asymptotic behaviour at infinity and the existence of solutions for every $n\ge 3$ also in a supercritical regime. Finally, we state some open problems.
We describe the asymptotic behavior of Palais-Smale sequences associated to certain Yamabe-type equations on manifolds with boundary. We prove that each of those sequences converges to a solution of the limit equation plus a finite number of "bubbles" which are obtained by rescaling fundamental solutions of the corresponding Euclidean equations.
In this paper, we prove the existence of hypersurfaces in the Euclidean space with prescribed boundary and whose k-th Weingarten curvature equals a given function that depends on the normal of the hypersurface. The proof is based on the solvability of a fully nonlinear elliptic PDE. The required a priori estimates are established under the natural assumptions that the prescribed boundary is strictly convex and the prescribed function satisfies a Serrin type condition.
In this paper we are concerned with the problem of finding hypersurfaces of constant curvature and prescribed boundary in the Euclidean space, using the theory of fully nonlinear elliptic equations. We prove that if the given data admits a suitable radial graph as a subsolution, then we can find a radial graph with constant curvature and that realizes the prescribed boundary. As an application we prove that if $Ω\subset\mathbb{S}^n$ is a mean convex domain whose closure is contained in an open hemisphere of $\mathbb{S}^n$ then, for $0<R<n(n-1),$ there exists a radial graph of constant scalar c…
We consider the class of all conformal mappings from a compact Riemann surface into the threedimensional or fourdimensional Euclidean space. A sequence in this class with bounded Willmore functional is shown to have a sequence of conformal transformations of the target space, such that a subsequence of the transformed sequence converges. This implies that there exists a smooth conformal mapping, which minimizes the Willmore functional in this class. For this purpose we extend the quaternionic function theory of Pedit and Pinkall to square integrable Hopf fields. In particular, we proof the Plu…
