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Jérôme Vétois

Publications and source records attributed to Jérôme Vétois.

At least 19 recordsLinked to original sources
arXiv2026

Nonexistence of minimizers for the second conformal eigenvalue near the round sphere in low dimensions

We consider the problem of minimizing the second conformal eigenvalue of the conformal Laplacian in a conformal class of metrics with renormalized volume. We prove, in dimensions $n\in\left\{3,\dotsc,10\right\}$, that a minimizer for this problem does not exist for metrics sufficiently close to the round metric on the sphere. This is in striking contrast with the situation in dimensions $n \ge 11$, where Ammann and Humbert obtained the existence of minimizers for the second conformal eigenvalue on any smooth closed non-locally conformally flat manifold. As a byproduct of our techniques, we also obtain a lower bound on the energy of sign-changing solutions of the \nobreak Yamabe equation in dimensions 3, 4 and 5, which extends a result obtained by Weth in the case of the round sphere.

math.DG↗
arXiv2026

Positivity and non-positivity results for the sixth-order $Q$-curvature of conformal metrics in $\mathbb{R}^n$

Given $n,m\in\mathbb{N}$ such that $n\ge2m\ge4$, letting $g$ be a conformally Euclidean metric on $\mathbb{R}^n$, we consider the question of positivity of the lower-order $Q$-curvatures $Q_g^{(2k)}$ for $k\in\left\{1,\dotsc,m-1\right\}$ when $Q_g^{(2m)}$ is assumed to be nonnegative and not identically zero. We assume moreover that the scalar curvature of the metric $g$ is nonnegative near infinity if $n=2m$ or that $Q_g^{(2m)}$ satisfies a slow decay barrier condition near infinity if $n>2m$. Positive results for this question have been obtained by Gursky and Malchiodi for $m=2$ and $k=1$ in the context of closed manifolds with nonnegative scalar curvature and by Li and Xu and Li, Wei, and Xu for $m\ge2$ and $k\in\left\{1,\min(m-1,2)\right\}$ in the context of conformally Euclidean metrics on $\mathbb{R}^n$. These results hold for all $n\ge2m$. Considering the case where $m\ge4$ and $k=3$, we obtain a positive result for this question when $n\in\left\{2m,2m+1,\dotsc,4m-6\right\}$, namely for these dimensions, we obtain that if $Q_g^{(2m)}\ge0$ and $Q_g^{(2m)}\not\equiv0$ in $\mathbb{R}^n$, then $Q_g^{(6)}>0$. On the other hand, in surprising contrast with the results of Gursky and Malchiodi, Li and Xu, and Li, Wei, and Xu, we find that the answer to this question is negative when $k=3$ and $n\ge N_m$ for some $N_m\in\mathbb{R}$. In this case, we are able to construct examples of conformally Euclidean metrics such that $Q_g^{(2m)}$ is positive everywhere, but $Q_g^{(6)}$ is negative at some point. By stereographic projection, our examples extend to metrics conformal to the standard metric on $\mathbb{S}^n$.

math.DG↗
arXiv2023

Liouville-type results for the CR Yamabe equation in the Heisenberg group

We obtain Liouville-type results for solutions to the CR Yamabe equation in $\mathbb{H}^n$, which extend a result obtained by Jerison and Lee for solutions in $L^{2+2/n}(\mathbb{H}^n)$. We obtain our results under either pointwise conditions or integral conditions at infinity. In particular, our results hold for all bounded solutions when $n=2$ and solutions satisfying a pointwise decay assumption when $n\ge3$. The proofs rely on integral estimates combined with a suitable divergence formula.

math.AP↗
arXiv2022

Existence results for the higher-order $Q$-curvature equation

We obtain existence results for the $Q$-curvature equation of order $2k$ on a closed Riemannian manifold of dimension $n\ge 2k+1$, where $k\ge1$ is an integer. We obtain these results under the assumptions that the Yamabe invariant of order $2k$ is positive and the Green's function of the corresponding operator is positive, which are satisfied for instance when the manifold is Einstein with positive scalar curvature. In the case where $2k+1\le n\le2k+3$ or $(M,g)$ is locally conformally flat, we assume moreover that the operator has positive mass. In the case where $n\ge2k+4$ and $(M,g)$ is not locally conformally flat, the results essentially reduce to the determination of the sign of a complicated constant depending only on $n$ and $k$.

math.AP↗
arXiv2022

Stability and instability results for sign-changing solutions to second-order critical elliptic equations

On a smooth, closed Riemannian manifold $\left(M,g\right)$ of dimension $n\ge3$, we consider the stationary Schrödinger equation $Δ_gu+h_0u=\left|u\right|^{2^*-2}u$, where $Δ_g:=-\text{div}_g\nabla$, $h_0\in C^1\left(M\right)$ and $2^* :=\frac{2n}{n-2}$. We prove that, up to perturbations of the potential function $h_0$ in $C^1\left(M\right)$, the sets of sign-changing solutions that are bounded in $H^1\left(M\right)$ are precompact in the $C^2$ topology. We obtain this result under the assumptions that $\left(M,g\right)$ is locally conformally flat, $n\ge7$ and $h_0\ne\frac{n-2}{4\left(n-1\right)}\text{Scal}_g$ at all points in $M$, where $\text{Scal}_g$ is the scalar curvature of the manifold. We then provide counterexamples in every dimension $n\ge3$ showing the optimality of these assumptions.

math.AP↗
arXiv2022

Sign-changing blow-up for the Yamabe equation at the lowest energy level

We investigate the blow-up behavior of sequences of sign-changing solutions for the Yamabe equation on a Riemannian manifold $(M,g)$ of positive Yamabe type. For each dimension $n\ge11$, we describe the value of the minimal energy threshold at which blow-up occurs. In dimensions $11 \le n \le 24$, where the set of positive solutions is known to be compact, we show that the set of sign-changing solutions is not compact and that blow-up already occurs at the lowest possible energy level. We prove this result by constructing a smooth, non-locally conformally flat metric on space forms $\mathbb{S}^n/Γ$, $Γ\neq \{1\}$, whose Yamabe equation admits a family of sign-changing blowing-up solutions. As a counterpart of this result, we also prove a sharp compactness result for sign-changing solutions at the lowest energy level, in small dimensions or under strong geometric assumptions.

math.AP↗
arXiv2021

Sign-changing blow-up for the Moser-Trudinger equation

Given a sufficiently symmetric domain $Ω\Subset\mathbb{R}^2$, for any $k\in \mathbb{N}\setminus \{0\}$ and $β>4πk$ we construct blowing-up solutions $(u_\varepsilon)\subset H^1_0(Ω)$ to the Moser-Trudinger equation such that as $\varepsilon\downarrow 0$, we have $\|\nabla u_\varepsilon\|_{L^2}^2\to β$, $u_\varepsilon \rightharpoonup u_0$ in $H^1_0$ where $u_0$ is a sign-changing solution of the Moser-Trudinger equation and $u_\varepsilon$ develops $k$ positive spherical bubbles, all concentrating at $0\in Ω$. These $3$ features (lack of quantization, non-zero weak limit and bubble clustering) stand in sharp contrast to the positive case ($u_\varepsilon>0$) studied by the second author and Druet (J. Eur. Math. Soc. (JEMS) 22 (2020)).

math.AP↗
arXiv2020

Sharp pointwise estimates for weighted critical $p$-Laplace equations

We investigate the asymptotic behavior of solutions to a class of weighted quasilinear elliptic equations which arise from the Euler--Lagrange equation associated with the Caffarelli--Kohn--Nirenberg inequality. We obtain sharp pointwise estimates which extend and improve previous results obtained in the unweighted case. In particular, we show that we can refine the asymptotic expansion at infinity by using a Kelvin-type transformation, which reduces the problem to another elliptic-type problem near the origin. The application of this transformation is straightforward in the linear case but more delicate in the quasilinear case. In particular, it is necessary in this case to establish some preliminary estimates before being able to apply the transformation.

math.AP↗
arXiv2019

Blowing-up solutions for second-order critical elliptic equations: the impact of the scalar curvature

Given a closed manifold $(M^n,g)$, $n\geq 3$, Olivier Druet proved that a necessary condition for the existence of energy-bounded blowing-up solutions to perturbations of the equation $$Δ_gu+h_0u=u^{\frac{n+2}{n-2}},\ u>0\hbox{ in }M$$ is that $h_0\in C^1(M)$ touches the Scalar curvature somewhere when $n\geq 4$ (the condition is different for $n=6$). In this paper, we prove that Druet's condition is also sufficient provided we add its natural differentiable version. For $n\geq 6$, our arguments are local. For the low dimensions $n\in\{4,5\}$, our proof requires the introduction of a suitable mass that is defined only where Druet's condition holds. This mass carries global information both on $h_0$ and $(M,g)$.

math.AP↗
arXiv2019

Examples of sharp asymptotic profiles of singular solutions to an elliptic equation with a sign-changing non-linearity

The first two authors [Proc. Lond. Math. Soc. (3) {\bf 114}(1):1--34, 2017] classified the behaviour near zero for all positive solutions of the perturbed elliptic equation with a critical Hardy--Sobolev growth $$-Δu=|x|^{-s} u^{2^\star(s)-1} -μu^q \hbox{ in }B\setminus\{0\},$$ where $B$ denotes the open unit ball centred at $0$ in $\mathbb{R}^n$ for $n\geq 3$, $s\in (0,2)$, $2^\star(s):=2(n-s)/(n-2)$, $μ>0$ and $q>1$. For $q\in (1,2^\star-1)$ with $2^\star=2n/(n-2)$, it was shown in the op. cit. that the positive solutions with a non-removable singularity at $0$ could exhibit up to three different singular profiles, although their existence was left open. In the present paper, we settle this question for all three singular profiles in the maximal possible range. As an important novelty for $μ>0$, we prove that for every $q\in (2^\star(s) -1,2^\star-1)$ there exist infinitely many positive solutions satisfying $|x|^{s/(q-2^\star(s)+1)}u(x)\to μ^{-1/(q-2^\star(s)+1)}$ as $|x|\to 0$, using a dynamical system approach. Moreover, we show that there exists a positive singular solution with $\liminf_{|x|\to 0} |x|^{(n-2)/2} u(x)=0$ and $\limsup_{|x|\to 0} |x|^{(n-2)/2} u(x)\in (0,\infty)$ if (and only if) $q\in (2^\star-2,2^\star-1)$.

math.AP↗
arXiv2019

Convergence result and blow-up examples for the Guan--Li mean curvature flow on warped product spaces

We examine the question of convergence of solutions to a geometric flow which was introduced by Guan and Li for starshaped hypersurfaces in space forms and generalized by Guan, Li, and Wang to the case of warped product spaces. We obtain a convergence result under a condition on the optimal modulus of continuity of the initial data. Moreover we show by examples that this condition is optimal at least in the one-dimensional case.

math.AP↗
arXiv2016

Positive clusters for smooth perturbations of a critical elliptic equation in dimensions four and five

We construct clustering positive solutions for a perturbed critical elliptic equation on a closed manifold of dimension $n=4,5$. Such a construction is already available in the literature in dimensions $n\ge 6$ (see for instance [8,12,27,29,33]) and not possible in dimension $3$ by [25]. This also provides new patterns for the Lin--Ni [21] problem on closed manifolds and completes results by Brézis and Li [6] about this problem.

math.AP↗
arXiv2015

Decay estimates and a vanishing phenomenon for the solutions of critical anisotropic equations

We investigate the asymptotic behavior of solutions of anisotropic equations of the form $-\sum_{i=1}^n\partial_{x_i}(\left|\partial_{x_i}u\right|^{p_i-2}\partial_{x_i}u)=f(x,u)$ in $\mathbb{R}^n$, where $p_i>1$ for all $i=1,\dotsc,n$ and $f$ is a Caratheodory function with critical Sobolev growth. This problem arises in particular from the study of extremal functions for a class of anisotropic Sobolev inequalities. We establish decay estimates for the solutions and their derivatives, and we bring to light a vanishing phenomenon which occurs when the maximum value of the exponents $p_i$ exceeds a critical value.

math.AP↗