arXiv subjects
Publications and source records attributed to Frédéric Robert.
We construct blowing-up sign-changing solutions to some nonlinear critical equations by glueing a standard bubble to a degenerate function. We develop a method based on analyticity to perform the glueing when the critical manifold of solutions is degenerate and no Bianchi--Egnell type condition holds.
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.
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.
We prove a pointwise control for the Green's function of polyharmonic operators with holes: this control is uniform while holes shrink. For the usual Laplacian, such a control is given by the maximum principle; the techniques developed here applies to general polyharmonic operators for which there is no comparison principle.
We prove some refined asymptotic estimates for postive blowing up solutions to $Δu+εu=n(n-2)u^{\frac{n+2}{n-2}}$ on $Ω$, $\partial_νu=0$ on $\partialΩ$; $Ω$ being a smooth bounded domain of $\rn$, $n\geq 3$. In particular, we show that concentration can occur only on boundary points with nonpositive mean curvature when $n=3$ or $n\geq 7$. As a direct consequence, we prove the validity of the Lin-Ni's conjecture in dimension $n=3$ and $n\geq 7$ for mean convex domains and with bounded energy. Recent examples by Wang-Wei-Yan show that the bound on the energy is a necessary condition.
Optimal pointwise estimates are derived for the biharmonic Green function under Dirichlet boundary conditions in arbitrary $C^{4,γ}$-smooth domains. Maximum principles do not exist for fourth order elliptic equations and the Green function may change sign. It prevents using a Harnack inequality as for second order problems and hence complicates the derivation of optimal estimates. The present estimate is obtained by an asymptotic analysis. The estimate shows that this Green function is positive near the singularity and that a possible negative part is small in the sense that it is bounded by the product of the squared distances to the boundary.
We investigate fourth order Paneitz equations of critical growth in the case of $n$-dimensional closed conformally flat manifolds, $n \ge 5$. Such equations arise from conformal geometry and are modelized on the Einstein case of the geometric equation describing the effects of conformal changes of metrics on the $Q$-curvature. We obtain sharp asymptotics for arbitrary bounded energy sequences of solutions of our equations from which we derive stability and compactness properties. In doing so we establish the criticality of the geometric equation with respect to the trace of its second order terms.
We give sufficient conditions on a function invariant under the action of an isometry group to be Branson's Q-curvature of a metric in a given conformal class, using the conformal GJMS operators.
In general, for higher order elliptic equations and boundary value problems like the biharmonic equation and the linear clamped plate boundary value problem neither a maximum principle nor a comparison principle or -- equivalently -- a positivity preserving property is available. The problem is rather involved since the clamped boundary conditions prevent the boundary value problem {from} being reasonably written as a system of second order boundary value problems. It is shown that, on the other hand, for bounded smooth domains $Ω\subset\mathbb{R}^n$, the negative part of the corresponding Green's function is "small" when compared with its singular positive part, provided $n\ge 3$. Moreover, the biharmonic Green's function in balls $B\subset\mathbb{R}^n$ under Dirichlet (i.e. clamped) boundary conditions is known explicitly and is positive. It has been known for some time that positivity is preserved under small regular perturbations of the domain, if $n=2$. In the present paper, such a stability result is proved for $n\ge 3$. Keywords: Biharmonic Green's functions, positivity, almost positivity, blow-up procedure.
Using the Mountain--Pass Theorem of Ambrosetti and Rabinowitz we prove that $-Δ_p u-μ|x|^{-p}{u^{p-1}}=|x|^{-s}{u^{\crits-1}}+u^{\crit-1}$ admits a positive weak solution in $\rn$ of class $\dunp\cap C^1(\rn\setminus\{0\})$, whenever $μ<μ_1$, and $μ_1=[(n-p)/p]^p$. The technique is based on the existence of extremals of some Hardy--Sobolev type embeddings of independent interest. We also show that if $u\in\dunp$ is a weak solution in $\rn$ of $-Δ_p u-μ|x|^{-p}{|u|^{p-2}u}=|x|^{-s}{|u|^{\crits-2}u}+|u|^{q-2}u$, then $u\equiv0$ when either $1 \crit$ and $u$ is also of class $L^\infty_\text{\scriptsize{loc}}(\rn\setminus\{0\})$.
We prove in this article that the local image of each conformal $Q$-curvature operator of arbitrary order on the sphere admits no scalar constraint. However, we prove that identities of Kazdan--Warner type hold for its graph.