arXiv subjects

Jérôme Vétois

Publications and source records attributed to Jérôme Vétois.

23 records · Page 2Linked to original sources

arXiv2014

A priori estimates and application to the symmetry of solutions for critical $p$-Laplace equations

We establish pointwise a priori estimates for solutions in $D^{1,p}(\mathbb{R}^n)$ of equations of type $-Δ_pu=f(x,u)$, where $p\in(1,n)$, $Δ_p:=\mbox{div}\big(\left|\nabla u\right|^{p-2}\nabla u\big)$ is the $p$-Laplace operator, and $f$ is a Caratheodory function with critical Sobolev growth. In the case of positive solutions, our estimates allow us to extend previous radial symmetry results. In particular, by combining our results and a result of Damascelli-Ramaswamy, we are able to extend a recent result of Damascelli-Merchán-Montoro-Sciunzi on the symmetry of positive solutions in $D^{1,p}(\mathbb{R}^n)$ of the equation $-Δ_pu=u^{p^*-1}$, where $p^*:=np/(n-p)$.

math.AP↗

arXiv2014

Examples of non-isolated blow-up for perturbations of the scalar curvature equation on non locally conformally flat manifolds

Solutions to scalar curvature equations have the property that all possible blow-up points are isolated, at least in low dimensions. This property is commonly used as the first step in the proofs of compactness. We show that this result becomes false for some arbitrarily small, smooth perturbations of the potential.

math.AP↗

arXiv2012

Blow-up solutions for linear perturbations of the Yamabe equation

For a smooth, compact Riemannian manifold (M,g) of dimension $N \geg 3$, we are interested in the critical equation $$Δ_g u+(N-2/4(N-1) S_g+εh)u=u^{N+2/N-2} in M, u>0 in M,$$ where Δ_g is the Laplace--Beltrami operator, S_g is the Scalar curvature of (M,g), $h\in C^{0,α}(M)$, and $ε$ is a small parameter.

math.AP↗

arXiv2012

The effect of linear perturbations on the Yamabe problem

In conformal geometry, the Compactness Conjecture asserts that the set of Yamabe metrics on a smooth, compact, aspherical Riemannian manifold (M,g) is compact. Established in the locally conformally flat case by Schoen [43,44] and for n\leq 24 by Khuri-Marques-Schoen [26], it has revealed to be generally false for n\geq 25 as shown by Brendle [8] and Brendle-Marques [9]. A stronger version of it, the compactness under perturbations of the Yamabe equation, is addressed here with respect to the linear geometric potential n-2/4(n-1) Scal_g, Scal_g being the Scalar curvature of (M,g). We show that a-priori L^\infty-bounds fail for linear perturbations on all manifolds with n\geq 4 as well as a-priori gradient L^2--bounds fail for non-locally conformally flat manifolds with n\geq 6 and for locally conformally flat manifolds with n\geq 7. In several situations, the results are optimal. Our proof combines a finite dimensional reduction and the construction of a suitable ansatz for the solutions generated by a family of varying metrics in the conformal class of g.

math.AP↗

arXiv2012

Sign-changing blow-up for scalar curvature type equations

Given $(M,g)$ a compact Riemannian manifold of dimension $n\geq 3$, we are interested in the existence of blowing-up sign-changing families $(\ue)_{\eps>0}\in C^{2,θ}(M)$, $θ\in (0,1)$, of solutions to $$Δ_g \ue+h\ue=|\ue|^{\frac{4}{n-2}-\eps}\ue\hbox{ in }M\,,$$ where $Δ_g:=-\hbox{div}_g(\nabla)$ and $h\in C^{0,θ}(M)$ is a potential. We prove that such families exist in two main cases: in small dimension $n\in \{3,4,5,6\}$ for any potential $h$ or in dimension $3\leq n\leq 9$ when $h\equiv\frac{n-2}{4(n-1)}\Scal_g$. These examples yield a complete panorama of the compactness/noncompactness of critical elliptic equations of scalar curvature type on compact manifolds. The changing of the sign is necessary due to the compactness results of Druet and Khuri--Marques--Schoen.

math.AP↗