Related papers: Solving the Yamabe Problem by an Iterative Method …

We introduce an iterative scheme to prove the Yamabe problem $ - a\Delta_{g} u + S u = \lambda u^{p-1} $, firstly on open domain $ (\Omega, g) $ with Dirichlet boundary conditions, and then on closed manifolds $ (M, g) $ by local argument.…

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 $,…

We apply iteration schemes and perturbation methods to provide a complete solution of the boundary Yamabe problem with minimal boundary scenario, or equivalently, the existence of a real, positive, smooth solution of $ -\frac{4(n -1)}{n -…

We start by taking the analytical approach to discuss how the minimizer of Yamabe functional provides constant scalar curvature and its relationship with the Sobolev Space $W^{1,2}.$ Then, after demonstrating the importance of the sphere…

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 $f$…

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…

We study positive solutions of the equation $-\Delta_g u + \lambda u = \lambda u^q$, with $\lambda >0$, $q>1$ on the round sphere $\mathbb{S}^n$ . We reduce the equation to an ordinary differential equation by considering isoparametric…

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 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 $…

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…

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…

We study the existence of solutions to the spinorial Yamabe equation -- that is, the Euler--Lagrange equation associated with the conformal invariant introduced by S. Raulot -- for compact manifolds with boundary. For the inhomogeneous…

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…

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…

Let $G=(V,E)$ be a locally finite graph, $\Omega\subset V$ be a bounded domain, $\Delta$ be the usual graph Laplacian, and $\lambda_1(\Omega)$ be the first eigenvalue of $-\Delta$ with respect to Dirichlet boundary condition. Using the…

This paper deals with the equation $-\Delta u+\mu u=f$ on high-dimensional spaces $\mathbb{R}^m$, where the right-hand side $f(x)=F(Tx)$ is composed of a separable function $F$ with an integrable Fourier transform on a space of a dimension…

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…

Let $(M, g)$ be a compact Riemannian manifold with boundary. The Yamabe problem concerning the existence of a metric conformally equivalent to $g$ having constant scalar curvature on $M$ and constant mean curvature on its boundary is…

We study the existence of non--trivial solutions to the Yamabe equation: $$-\Delta u+ a(x)= \mu u|u|^\frac4{n-2} \hbox{} \mu >0, x\in \Omega \subset {\mathbf R}^n \hbox{with} n\geq 4,$$ $$ u(x)=0 \hbox{on} \partial \Omega$$ under weak…

We study Yamabe-type equations on the product of two spheres $(S^n \times S^n, G_\delta)$, where $G_\delta$ is a family of Riemannian metrics parametrized by $\delta > 0$. Using bifurcation theory and isoparametric functions, we establish…