Related papers: Solutions of the Yamabe Equation By Lyapunov-Schmi…

Given any closed Riemannian manifold $(M, g)$, we use the gradient flow method and Sign-Changing Critical Point Theory to prove multiplicity results for 2-nodal solutions of a subcritical Yamabe type equation on $(M, g)$. If $(N, h)$ is 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 $…

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…

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 $(M^m,g)$ be a closed Riemannian manifold $(m\geq 2)$ of positive scalar curvature and $(N^n,h)$ any closed manifold. We study the asymptotic behaviour of the second Yamabe constant and the second $N-$Yamabe constant of $(M\times…

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…

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…

Let $(M,g)$ be a compact Riemannian manifold of dimension $n\geq 3$. Under some assumptions, we prove that there exists a positive function $\varphi$ solution of the following Yamabe type equation \Delta \varphi+ h\varphi= \tilde h…

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 $f$…

For a closed Riemannian manifold $(M^m,g)$ of constant positive scalar curvature and any other closed Riemannian manifold $(N^n,h)$, we show that the limit of the Yamabe constants of the Riemannian products $(M\times N,g+rh)$ as $r$ goes to…

We prove that certain subcritical Paneitz-Branson type equations on a closed Riemannian manifold $(M,g)$ have at least $\mathrm{Cat}(M)$ positIve solutions, where $\mathrm{Cat}(M)$ is the Lusternik-Schnirelmann category of $M$. This implies…

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 =…

We introduce a double iterative scheme and local variational method to solve the Yamabe-type equation $ - \frac{4(n - 1)}{n - 2}\Delta_{g} u + (S_{g} + \beta ) u = \lambda u^{\frac{n + 2}{n - 2}} $ for some constant $ \beta \leqslant 0 $,…

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…

We introduce an iterative scheme to solve the Yamabe equation $ - a\Delta_{g} u + S u = \lambda u^{p-1} $ on small domains $(\Omega,g)\subset {\mathbb R}^n$ equipped with a Riemannian metric $g$. Thus $g$ admits a conformal change to a…

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…

We establish the existence of infinitely many complete metrics with constant scalar curvature on prescribed conformal classes on certain noncompact product manifolds. These include products of closed manifolds with constant positive scalar…

Let $(M^n, g)$ and $(X^m, h)$ be closed manifolds $m, n>2$, such that $(X, h)$ has constant positive scalar curvature. We consider the one parameter family of products $(M\times X, g+\epsilon^2 h)$, $\epsilon>0$. We prove that if either the…

We prove the existence of a solution of the Yamabe equation on complete manifolds with finite volume and positive Yamabe invariant. In order to circumvent the standard methods on closed manifolds which heavily rely on global (compact)…