Searcharxiv⌕ Search

arXiv subjects

Frédéric Robert

Publications and source records attributed to Frédéric Robert.

At least 19 recordsLinked to original sources
arXiv2026

Beyond the analytic solution manufacturing for anisotropic elliptic equations

This article is aimed at introducing a new procedure to define reference solutions to which the numerical approximations of anisotropic problems arising in plasma physics may be compared. In this context, the main issue is related to the construction of functions with no gradients along the vector field defining the anisotropy direction. The so-called solution manufacturing, implemented so far in the literature, only applies to frame- work where all the computations can be carried out analytically. In the present paper, we propose a new method in which this requirement is not mandatory. Indeed, via a Riemannian metric, we use a propagation of the boundary value by the flows related to the vector field defining the anisotropy direction. This permits to extend the application of the solution manufacturing to a broader range of contexts.

math.NA↗
arXiv2025

Sharp multiscale control for high order nonlinear equations

We analyze the behavior of families $(u_α)_{α>0}$ of solutions to the high-order critical equation $P_αu_α=Δ_g^k u_α+\hbox{lot}=|u_α|^{2^\star-2}u_α$ on a Riemannian manifold $M$, with a uniform bound on the Dirichlet energy. We prove a sharp pointwise control of the $u_α$'s by a sum of bubbles uniformly with respect to $α\to +\infty$, that is $|u_α|\leq C\Vert u_\infty \Vert_\infty +C\sum_{i=1}^NB_{i,α}$ where $u_\infty \in C^{2k}(M)$ and the $(B_{i,α})_α$, $i=1,...,N$ are explicit standard peaks.

math.AP↗
arXiv2025

Ground-state blowing-up solutions for a Hardy-Sobolev equations on a manifold

We prove the existence of blowing-up families of solutions to an equation of Hardy-Sobolev type in high dimensions. These families are of minimal type. The sole condition is that the potential of the linear operator touches a critical potential at the singular point. This condition is sharp as shown by the first author in [Cheikh-Ali, Pacific J. of Math. 2022].

math.AP↗
arXiv2025

Optimal Sobolev inequalities of high order with $L^2$-remainder

We investigate the validity of the optimal higher-order Sobolev inequality $H_k^2(M^n)\hookrightarrow L^{\frac{2n}{n-2k}}(M^n)$ on a closed Riemannian manifold when the remainder term is the $L^2-$norm. Unlike the case $k=1$, the optimal inequality does not hold in general for $k>1$. We prove conditions for the validity and non-validity that depend on the geometry of the manifold. Our conditions are sharp when $k=2$ and in small dimensions.

math.AP↗
arXiv2025

Concentration analysis for elliptic critical equations with no boundary control: ground-state blow-up

We perform the apriori analysis of solutions to critical nonlinear elliptic equations on manifolds with boundary. The solutions are of minimizing type. The originality is that we impose no condition on the boundary, which leads us to assume $L^2-$concentration. We also analyze the effect of a non-homogeneous nonlinearity that results in the fast convergence of the concentration point.

math.AP↗
arXiv2024

Localization of bubbling for high order nonlinear equations

We analyze the asymptotic pointwise behavior of families of solutions to the high-order critical equation $$P_αu_α=Δ_g^k u_α+\hbox{lot}=|u_α|^{2^\star-2-ε_α} u_α\hbox{ in }M$$ that behave like $$u_α=u_0+B_α+o(1)\hbox{ in }H_k^2(M)$$ where $B=(B_α)_α$ is a Bubble, also called a Peak. We give obstructions for such a concentration to occur: depending on the dimension, they involve the mass of the associated Green's function or the difference between $P_α$ and the conformally invariant GJMS operator. The bulk of this analysis is the proof of the pointwise control \begin{equation*} |u_α(x)|\leq C\Vert u_0\Vert_\infty^{(2^\star-1)^2}+C\left(\frac{μ_α^{2}}{μ_α^{2 }+d_g(x,x_α)^{2 }}\right)^{\frac{n-2k}{2}}\hbox{ for all }x\in M\hbox{ and }α\in\mathbb{N}, \end{equation*} where $|u_α(x_α)|=\max_M|u_α|\to +\infty$ and $μ_α:=|u_α(x_α)|^{-\frac{2}{n-2k}}$. The key to obtain this estimate is a sharp control of the Green's function for elliptic operators involving a Hardy potential.

math.AP↗
arXiv2024

One-bubble nodal blow-up for asymptotically critical stationary Schrödinger-type equations

We investigate in this work families $(u_ε)_{ε>0}$ of sign-changing blowing-up solutions of asymptotically critical stationary nonlinear Schrödinger equations of the following type: $$Δ_g u_ε+ h_εu_ε= |u_ε|^{p_ε-2} u_ε$$ in a closed manifold $(M,g)$, where $h_ε$ converges to $h$ in $C^1(M)$. Assuming that $(u_ε)_{ε>0}$ blows-up as \emph{a single sign-changing bubble}, we obtain necessary conditions for blow-up that constrain the localisation of blow-up points and exhibit a strong interaction between $h$, the geometry of $(M,g)$ and the bubble itself. These conditions are new and are a consequence of the sign-changing nature of $u_ε$.

math.AP↗
arXiv2021

The Hardy--Schrödinger Operator on the Poincaré Ball: Compactness and Multiplicity

Let $Ω$ be a compact smooth domain containing zero in the Poincaré ball model of the Hyperbolic space $\mathbb{B}^{n}$ ($n \geq 3$) and let $-Δ_{\mathbb{B}^{n}}$ be the Laplace-Beltrami operator on $\mathbb{B}^{n}$, associated with the metric $g_{\mathbb{B}^{n}}= \frac{4}{(1-|x|^{2})^2}g_{_{\hbox{Eucl}}}$. We consider issues of non-existence, existence, and multiplicity of variational solutions for the borderline Dirichlet problem, \begin{eqnarray*} (E)~ \left\{ \begin{array}{lll} -Δ_{\mathbb{B}^{n}}u-γ{V_2}u -λu&=V_{2^\star(s)}|u|^{2^\star(s)-2}u &\hbox{ in }Ω\\ \hfill u &=0 & \hbox{ on } \partial Ω, \end{array} \right. \end{eqnarray*} where $0\leq γ\leq \frac{(n-2)^2}{4}$, $0< s <2$, ${2^\star(s)}:=\frac{2(n-s)}{n-2}$ is the corresponding critical Sobolev exponent, $V_{2}$ (resp., $V_{2^\star(s)}$) is a Hardy-type potential (resp., Hardy-Sobolev weight) that is invariant under hyperbolic scaling and which behaves like $\frac{1}{r^{2}}$ (resp., $\frac{1}{r^{s}}$) at the origin. The bulk of this paper is a sharp blow-up analysis on approximate solutions of $(E)$ with bounded but arbitrary high energies. Our analysis leads to existence of positive ground state solutions for $(E)$, whenever $n \geq 4$, $0 \leq γ\leq \frac{(n-2)^2}{4}-1$ and $ λ> 0$. The latter result also holds true for $n\geq 3$ and $γ> \frac{(n-2)^2}{4}-1$ provided the domain has a positive "hyperbolic mass". On the other hand, the same analysis yields that if $γ> \frac{(n-2)^2}{4}-1$ and the mass is non vanishing, then there is a surprising stability of regimes where no variational positive solution exists. As for higher energy solutions to $(E)$, we show that there are infinitely many of them provided $n\geq 5$, $0\leq γ<\frac{(n-2)^2}{4}-4$ and $ λ> \frac{n-2}{n-4} \left(\frac{n(n-4)}{4}-γ\right)$.

math.AP↗
arXiv2020

Multiplicity and stability of the Pohozaev obstruction for Hardy-Schrödinger equations with boundary singularity

Let $Ω$ be a smooth bounded domain in $\mathbb{R}^n$ ($n\geq 3$) such that $0\in\partial Ω$. In this memoir, we consider issues of non-existence, existence, and multiplicity of variational solutions in $H_{1,0}^2(Ω)$ for the borderline Dirichlet problem, $-Δu-γ\frac{u}{|x|^2}- h(x) u = \frac{|u|^{{2^\star(s)}-2}u}{|x|^s}$ in $Ω$, where $0<s<2$, ${2^\star(s)}:=\frac{2(n-s)}{n-2}$, $γ\in\mathbb{R}$ and $h\in C^0(\overlineΩ)$. We use sharp blow-up analysis on --possibly high energy-- solutions of corresponding subcritical problems to establish, for example, that if $γ<\frac{n^2}{4}-1$ and the principal curvatures of $\partialΩ$ at $0$ are non-positive but not all of them vanishing, then the above equation has an infinite number of (possibly sign-changing) solutions in ${H_{1,0}^2(Ω)}$. This complements results of the first and third authors, who had previously shown that if $γ\leq \frac{n^2}{4}-\frac{1}{4}$ and the mean curvature of $\partialΩ$ at $0$ is negative, then the equation has a positive solution. On the other hand, the sharp blow-up analysis also allows us to prove that if the mean curvature at $0$ is non-zero and if the mass (when defined) does not vanish, then there is a surprising stability under $C^1$-perturbations of the potential $h$ of those regimes where no variational positive solutions exist. In particular, and in sharp contrast with the non-singular case (i.e., when $γ=s=0$), we show non-existence of such solutions for (E) in any dimension, whenever $Ω$ is star-shaped and $h$ is close to $0$, which include situations not covered by the classical Pohozaev obstruction.

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↗
arXiv2017

The Hardy-Schrödinger operator with interior singularity: The remaining cases

We consider the remaining unsettled cases in the problem of existence of energy minimizing solutions for the Dirichlet value problem $L_γu-λu=\frac{u^{2^*(s)-1}}{|x|^s}$ on a smooth bounded domain $Ω$ in $\mathbb{R}^n$ ($n\geq 3$) having the singularity $0$ in its interior. Here $γ<\frac{(n-2)^2}{4}$, $0\leq s <2$, $2^*(s):=\frac{2(n-s)}{n-2}$ and $0\leq λ<λ_1(L_γ)$, the latter being the first eigenvalue of the Hardy-Schrödinger operator $L_γ:=-Δ-\fracγ{|x|^2}$. There is a threshold $λ^*(γ, Ω) \geq 0$ beyond which the minimal energy is achieved, but below which, it is not. It is well known that $λ^*(Ω) = 0$ in higher dimensions, for example if $0\leq γ\leq \frac{(n-2)^2}{4}-1$. Our main objective in this paper is to show that this threshold is strictly positive in "lower dimensions" such as when $ \frac{(n-2)^2}{4}-1<γ<\frac{(n-2)^2}{4}$, to identify the critical dimensions (i.e., when the situation changes), and to characterize it in terms of $Ω$ and $γ$. If either $s>0$ or if $γ> 0$, i.e., in {\it the truly singular case}, we show that in low dimensions, a solution is guaranteed by the positivity of the "Hardy-singular internal mass" of $Ω$, a notion that we introduce herein. On the other hand, and just like the case wnen $γ=s=0$ studied by Brezis-Nirenberg and completed by Druet, $n=3$ is the critical dimension, and the classical positive mass theorem is sufficient for the {\it merely singular case}, that is when $s=0$, $γ\leq 0$.

math.AP↗
arXiv2017

Mass and Asymptotics associated to Fractional Hardy-Schrödinger Operators in Critical Regimes

We consider linear and non-linear boundary value problems associated to the fractional Hardy-Schrödinger operator $ L_{γ,α}: = ({-}{ Δ})^{\fracα{2}}- \fracγ{|x|^α}$ on domains of $\mathbb{R}^n$ containing the singularity $0$, where $0<α<2$ and $ 0 \le γ< γ_H(α)$, the latter being the best constant in the fractional Hardy inequality on $\mathbb{R}^n$. We tackle the existence of least-energy solutions for the borderline boundary value problem $(L_{γ,α}-λI)u= {\frac{u^{2^\star_α(s)-1}}{|x|^s}}$ on $Ω$, where $0\leq s <α<n$ and $ 2^\star_α(s)={\frac{2(n-s)}{n-α}}$ is the critical fractional Sobolev exponent. We show that if $γ$ is below a certain threshold $γ_{crit}$, then such solutions exist for all $0<λ<λ_1(L_{γ,α})$, the latter being the first eigenvalue of $L_{γ,α}$. On the other hand, for $γ_{crit}<γ<γ_H(α)$, we prove existence of such solutions only for those $λ$ in $(0, λ_1(L_{γ,α}))$ for which the domain $Ω$ has a positive {\it fractional Hardy-Schrödinger mass} $m_{γ, λ}(Ω)$. This latter notion is introduced by way of an invariant of the linear equation $(L_{γ,α}-λI)u=0$ on $Ω$.

math.AP↗
arXiv2016

Sharp asymptotic profiles for singular solutions to an elliptic equation with a sign-changing nonlinearity

Given $B_1(0)$ the unit ball of $\mathbb{R}^n$ ($n\geq 3$), we study smooth positive singular solutions $u\in C^2(B_1(0)\setminus \{0\})$ to $-Δu=\frac{u^{2^\star(s)-1}}{|x|^s}-μu^q$. Here $0< s<2$, $2^\star(s):=2(n-s)/(n-2)$ is critical for Sobolev embeddings, $q>1$ and $μ> 0$. When $μ=0$ and $s=0$, the profile at the singularity $0$ was fully described by Caffarelli-Gidas-Spruck. We prove that when $μ>0$ and $s>0$, besides this profile, two new profiles might occur. We provide a full description of all the singular profiles. Special attention is accorded to solutions such that $\liminf_{x\to 0}|x|^{\frac{n-2}{2}}u(x)=0$ and $\limsup_{x\to 0}|x|^{\frac{n-2}{2}}u(x)\in (0,+\infty)$. The particular case $q=(n+2)/(n-2)$ requires a separate analysis which we also perform.

math.AP↗
arXiv2015

Sobolev inequalities for the Hardy-Schrödinger operator: Extremals and critical dimensions

In this expository paper, we consider the Hardy-Schrödinger operator $-Δ-γ/|x|^2$ on a smooth domain Ωof R^n with 0\in\barΩ, and describe how the location of the singularity 0, be it in the interior of Ωor on its boundary, affects its analytical properties. We compare the two settings by considering the optimal Hardy, Sobolev, and the Caffarelli-Kohn-Nirenberg inequalities. The latter rewrites: $C(\int_Ω\frac{u^{p}}{|x|^s}dx)^{\frac{2}{p}}\leq \int_Ω |\nabla u|^2dx-γ\int_Ω\frac{u^2}{|x|^2}dx$ for all $u\in H^1_0(Ω)$, where γ<n^2/4, s\in [0,2) and p:=2(n-s)/(n-2). We address questions regarding the explicit values of the optimal constant C, as well as the existence of non-trivial extremals attached to these inequalities. Scale invariance properties lead to situations where the best constants do not depend on the domain and are not attainable. We consider two different approaches to "break the homogeneity" of the problem: One approach was initiated by Brezis-Nirenberg and by Janelli. It is suitable for the case where 0 is in the interior of Ω, and consists of considering lower order perturbations of the critical nonlinearity. The other approach was initiated by Ghoussoub-Kang , C.S. Lin et al. and Ghoussoub-Robert. It consists of considering domains where the singularity is on the boundary. Both of these approaches are rich in structure and in challenging problems. If 0\in Ω, a negative linear perturbation suffices for higher dimensions, while a positive "Hardy-singular interior mass" is required in lower dimensions. If the singularity is on the boundary, then the local geometry around 0 plays a crucial role in high dimensions, while a positive "Hardy-singular boundary mass" is needed for the lower dimensions.

math.AP↗
arXiv2014

On the Hardy-Schrödinger operator with a boundary singularity

We investigate the Hardy-Schrödinger operator $L_γ=-Δ-\fracγ{|x|^2}$ on domains $Ω\subset\rn$, whose boundary contain the singularity $0$. The situation is quite different from the well-studied case when $0$ is in the interior of $Ω$. For one, if $0\inΩ$, then $L_γ$ is positive if and only if $γ<\frac{(n-2)^2}{4}$, while if $0\in\partialΩ$ the operator $L_γ$ could be positive for larger value of $γ$, potentially reaching the maximal constant $\frac{n^2}{4}$ on convex domains. We prove optimal regularity and a Hopf-type Lemma for variational solutions of corresponding linear Dirichlet boundary value problems of the form $L_γ u=a(x)u$, but also for non-linear equations including $L_{_γ} u=\frac{|u|^{\crits-2}u}{|x|^s}$, where $γ<\frac{n^2}{4}$, $s\in [0,2)$ and $\crits:=\frac{2(n-s)}{n-2}$ is the critical Hardy-Sobolev exponent. We also provide a Harnack inequality and a complete description of the profile of all positive solutions --variational or not-- of the corresponding linear equation on the punctured domain. The value $γ=\frac{n^2-1}{4}$ turned out to be another critical threshold for the operator $L_γ$, and our analysis yields a corresponding notion of "Hardy singular boundary-mass" $m_γ(Ω)$ of a domain $Ω$ having $0\in \partial Ω$, which could be defined whenever $\frac{n^2-1}{4}<γ<\frac{n^2}{4}$.

math.AP↗