Related papers: The Yamabe problem with singularities

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…

Let $(M,g)$ be a compact Riemannian manifold with non-empty boundary. Provided $f$ an isoparametric function of $(M,g)$ we prove existence results for positive solutions of the Yamabe equation that are constant along the level sets of $f$.…

We consider a closed Riemannian manifold $(M^n ,g)$ of dimension $n\geq 3$ and study positive solutions of the equation $-\Delta_g u + \lambda u = \lambda u^q$, with $\lambda >0$, $q>1$. If $M$ supports a proper isoparametric function with…

Assume $\alpha\geq p>1$. Consider the following $p$-th Yamabe equation on a connected finite graph $G$: $$\Delta_p\varphi+h\varphi^{p-1}=\lambda f\varphi^{\alpha-1},$$ where $\Delta_p$ is the discrete $p$-Laplacian, $h$ and $f>0$ are fixed…

We consider on a closed Riemannian spin manifold $(M^n,g,\sigma)$ the spinorial Yamabe type equation $D_g\varphi=\lambda|\varphi|^{\frac{2}{n-1}}\varphi$, where $\varphi$ is a spinor field and $\lambda$ is a positive constant. For a…

Let $(M,g)$ be any closed Riemannianan manifold and $(N,h)$ be a Riemannian manifold of constant positive scalar curvature. We prove that the Yamabe equation on the Riemannian product $(M\times N , g + \delta h)$ has at least $Cat(M) +1 $…

In the first part of this thesis, we study the Yamabe problem with singularities, that we can announce as follow: Given a compact Riemannian manifold $(M,g)$, find a constant scalar curvature metric, conformal to $g$, when $g$ has not…

Given any closed Riemannian manifold $(M,g)$ we use the Lyapunov-Schmidt finite-dimensional reduction method and the classical Morse and Lusternick-Schnirelmann theories to prove multiplicity results for positive solutions of a subcritical…

On a closed Riemannian manifold $(M^n ,g)$, we consider the Yamabe-type equation $-\Delta_g u + \lambda u = \lambda |u|^{q-1}u$, where $\lambda \in \mathbb{R}_{+}$ and $q>1$. We assume that $M$ admits a proper isoparametric function $f$…

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 $-\Delta_g u + \lambda u =…

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…

Let $G=(V,E)$ be a connected infinite and locally finite weighted graph, $\Delta_p$ be the $p$-th discrete graph Laplacian. In this paper, we consider the $p$-th Yamabe type equation $$-\Delta_pu+h|u|^{p-2}u=gu^{\alpha-1}$$ on $G$, where…

Let $(M,g)$ be a closed Riemannian manifold of dimension $n\geq 3$ and $x_0 \in M$ be an isolated local minimum of the scalar curvature $s_g$ of $g$. For any positive integer $k$ we prove that for $\epsilon >0$ small enough the subcritical…

In this paper, we set up a new Yamabe type flow on a compact Riemannian manifold $(M,g)$ of dimension $n\geq 3$. Let $\psi(x)$ be any smooth function on $M$. Let $p=\frac{n+2}{n-2}$ and $c_n=\frac{4(n-1)}{n-2}$. We study the Yamabe-type…

We study the existence of conformal metrics on non-compact Riemannian manifolds with non-compact boundary, which are complete as metric spaces and have negative constant scalar curvature in the interior and negative constant mean curvature…

In this paper, we investigate the prescribed scalar curvature problem on a non-compact Riemannian manifold $(M, \langle \, , \, \rangle)$, namely the existence of a conformal deformation of the metric $\langle \, , \, \rangle$ realizing a…

We consider the Yamabe equation on a complete non-compact Riemannian manifold and study the condition of stability of solutions. If $(M^m,g)$ is a closed manifold of constant positive scalar curvature, which we normalize to be $m(m-1)$, we…

For a smooth, compact Riemannian manifold (M,g) of dimension $N \geg 3$, we are interested in the critical equation $$\Delta_g u+(N-2/4(N-1) S_g+\epsilon h)u=u^{N+2/N-2} in M, u>0 in M,$$ where \Delta_g is the Laplace--Beltrami operator,…

Let $(M, g)$ be a closed Riemannian manifold of dimension $n \geq 3$, and let $h \in C^1(M)$ be such that the operator $\Delta_g + h$ is coercive. Fix $x_0 \in M$ and $s \in (0, 2)$. We obtain uniform bounds on the solutions of the critical…

We show that if a compact connected $n$-dimensional manifold $M$ has a conformal class containing two non-homothetic metrics $g$ and $\tilde g=e^{2\varphi}g$ with non-generic holonomy, then after passing to a finite covering, either $n=4$…