The Klein–Gordon equation in the Anti-de Sitter cosmology

This paper deals with the Klein–Gordon equation on the Poincaré chart of the 5-dimensional Anti-de Sitter universe. When the mass μ is larger than $-\frac{1}{4}$, the Cauchy problem is well-posed despite the loss of global hyperbolicity due to the time-like horizon. We express the finite energy solutions in the form of a continuous Kaluza–Klein tower and we deduce a uniform decay as ${|t|}^{-\frac{3}{2}}$. We investigate the case $\mu =\frac{{\nu }^2-1}{2}$, $\nu \in {\mathbb{N}}^{⁎}$, which encompasses the gravitational fluctuations, $\nu =4$, and the electromagnetic waves, $\nu =2$. The propagation of the wave front set shows that the horizon acts like a perfect mirror. We establish that the smooth solutions decay as ${|t|}^{-2-\sqrt{\mu +\frac{1}{4}}}$, and we get global $L^p$ estimates of Strichartz type. When ν is even, there appears a lacuna and the equipartition of the energy occurs at finite time for the compactly supported initial data, although the Huygens principle fails. We address the cosmological model of the negative-tension Minkowski brane, on which a Robin boundary condition is imposed. We prove the hyperbolic mixed problem is well-posed and the normalizable solutions can be expanded into a discrete Kaluza–Klein tower. We establish some $L^2-L^{\infty }$ estimates in suitable weighted Sobolev spaces.     

Construction of solutions via local Pohozaev identities

This paper deals with the following nonlinear elliptic equation

$?\mathrm{\Delta }u+V(|y^{\prime }|,y^{″})u=u^{\frac{N+2}{N?2}},u>0,u\in H^1({\mathbb{R}}^N),$

where $(y^{\prime },y^{″})\in {\mathbb{R}}^2\times {\mathbb{R}}^{N?2}$, $V(|y^{\prime }|,y^{″})$ is a bounded non-negative function in ${\mathbb{R}}^{+}\times {\mathbb{R}}^{N?2}$. By combining a finite reduction argument and local Pohozaev type of identities, we prove that if $N\ge 5$ and $r^{2}V(r,y^{″})$ has a stable critical point $(r_0,y_0^{″})$ with $r_0>0$ and $V(r_0,y_0^{″})>0$, then the above problem has infinitely many solutions. This paper overcomes the difficulty appearing in using the standard reduction method to locate the concentrating points of the solutions.     

A proof of Wright's conjecture

Wright's conjecture states that the origin is the global attractor for the delay differential equation $y^{\prime }(t)=?\alpha y(t?1)[1+y(t)]$ for all $\alpha \in (0,\frac{\pi }{2}]$ when $y(t)>?1$. This has been proven to be true for a subset of parameter values α. We extend the result to the full parameter range $\alpha \in (0,\frac{\pi }{2}]$, and thus prove Wright's conjecture to be true. Our approach relies on a careful investigation of the neighborhood of the Hopf bifurcation occurring at $\alpha =\frac{\pi }{2}$. This analysis fills the gap left by complementary work on Wright's conjecture, which covers parameter values further away from the bifurcation point. Furthermore, we show that the branch of (slowly oscillating) periodic orbits originating from this Hopf bifurcation does not have any subsequent bifurcations (and in particular no folds) for $\alpha \in (\frac{\pi }{2},\frac{\pi }{2}+6.830\times {10}^{?3}]$. When combined with other results, this proves that the branch of slowly oscillating solutions that originates from the Hopf bifurcation at $\alpha =\frac{\pi }{2}$ is globally parametrized by $\alpha >\frac{\pi }{2}$.     

Existence of standing waves of nonlinear Schrödinger equations with potentials vanishing at infinity

For a singularly perturbed nonlinear elliptic equation ${\epsilon }^2\mathrm{\Delta }u?V(x)u+u^p=0$, $x\in {\mathbb{R}}^N$, we prove the existence of bump solutions concentrating around positive critical points of V when nonnegative V is not identically zero for $p\in (\frac{N}{N?2},\frac{N+2}{N?2})$ or nonnegative V satisfies ${\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{n}\mathrm{f}}_{|x|\to \infty }V(x){|x|}^2\log |x|>0$ for $p=\frac{N}{N?2}$.     

On logarithmic Sobolev inequalities for higher order fractional derivatives

On ${\mathbb{R}}^n$, we prove the existence of sharp logarithmic Sobolev inequalities with higher fractional derivatives. Let s be a positive real number. Any function f ∈ $H^s({\mathbb{R}}^n)$ satisfies

$\underset{{\mathbb{R}}^n}{\int }|f(x){|}^2\ln (\frac{{|f(x)|}^2}{6f{6}_2^2})\mathrm{d}x+(n+\frac{n}{s}\ln \alpha +\ln \frac{s\mathrm{\Gamma }(\frac{n}{2})}{\mathrm{\Gamma }(\frac{n}{2s})})6f{6}_2^2?\frac{{\alpha }^2}{{\pi }^s}6{(?\mathrm{\Delta })}^{s/2}f{6}_2^2$

with $\alpha >0$ be any number and where the operators ${(?\mathrm{\Delta })}^{s/2}$ in Fourier spaces are defined by $\stackrel{?}{{(?\mathrm{\Delta })}^{s/2}f}(k)\text{:}={(2\pi |k|)}^s\stackrel{?}{f}(k)$. To cite this article: A. Cotsiolis, N.K. Tavoularis, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Corrigendum to “Scattering above energy norm of solutions of a loglog energy-supercritical Schrödinger equation with radial data”

The purpose of this corrigendum is to point out some errors that appear in [1]. Our main result remains valid, i.e scattering of ${\stackrel{?}{H}}^k:={\dot{H}}^k({\mathbb{R}}^n)\cap {\dot{H}}^1({\mathbb{R}}^n)$ solutions of the loglog energy-supercritical Schrödinger equation $i{?}_{t}u+△u=|u{|}^{\frac{4}{n?2}}u{\log }^c?(\log ?(10+|u{|}^2)$, $0<c<c_n$, $n\in \{3,4\}$, with $k>\frac{n}{2}$, radial data $u(0):=u_0\in {\stackrel{?}{H}}^k$ but with slightly different values of $c_n$, i.e $c_n=\frac{1}{5772}$ if $n=3$ and $c_n=\frac{3}{8024}$ if $n=4$. We propose some corrections.     

Maximal linear groups induced on the Frattini quotient of a p-group

Let $p>3$ be a prime. For each maximal subgroup $H?\mathrm{GL}(d,p)$ with $|H|?p^{3d+1}$, we construct a d-generator finite p-group G with the property that $\mathrm{Aut}(G)$ induces H on the Frattini quotient $G/\mathrm{\Phi }(G)$ and $|G|?p^{\frac{d^4}{2}}$. A significant feature of this construction is that $|G|$ is very small compared to $|H|$, shedding new light upon a celebrated result of Bryant and Kovács. The groups G that we exhibit have exponent p, and of all such groups G with the desired action of H on $G/\mathrm{\Phi }(G)$, the construction yields groups with smallest nilpotency class, and in most cases, the smallest order.     

Odd symmetry of least energy nodal solutions for the Choquard equation

We consider the Choquard equation (also known as the stationary Hartree equation or Schrödinger–Newton equation)

$?\mathrm{\Delta }u+u=(I_{\alpha }?|u{|}^p)|u{|}^{p?2}u.$

Here $I_{\alpha }$ stands for the Riesz potential of order $\alpha \in (0,N)$, and $\frac{N?2}{N+\alpha }<\frac{1}{p}\le \frac{1}{2}$. We prove that least energy nodal solutions have an odd symmetry with respect to a hyperplane when α is either close to 0 or close to N.