Entire solutions for a semilinear fourth order elliptic problem with exponential nonlinearity

We investigate entire radial solutions of the semilinear biharmonic equation Δ²u=λexp(u) in Rn, n?5, λ>0 being a parameter. We show that singular radial solutions of the corresponding Dirichlet problem in the unit ball cannot be extended as solutions of the equation to the whole of Rn. In particular, they cannot be expanded as power series in the natural variable s=log|x|. Next, we prove the existence of infinitely many entire regular radial solutions. They all diverge to −∞ as |x|→∞ and we specify their asymptotic behaviour. As in the case with power-type nonlinearities [F. Gazzola, H.-Ch. Grunau, Radial entire solutions for supercritical biharmonic equations, Math. Ann. 334 (2006) 905-936], the entire singular solution x?−4log|x| plays the role of a separatrix in the bifurcation picture. Finally, a technique for the computer assisted study of a broad class of equations is developed. It is applied to obtain a computer assisted proof of the underlying dynamical behaviour for the bifurcation diagram of a corresponding autonomous system of ODEs, in the case n=5.     

Overdetermined problems with possibly degenerate ellipticity, a geometric approach

Given an open bounded connected subset Ω of ℝⁿ, we consider the overdetermined boundary value problem obtained by adding both zero Dirichlet and constant Neumann boundary data to the elliptic equation −div(A(|∇u|)∇u)=1 in Ω. We prove that, if this problem admits a solution in a suitable weak sense, then Ω is a ball. This is obtained under fairly general assumptions on Ω and A. In particular, A may be degenerate and no growth condition is required. Our method of proof is quite simple. It relies on a maximum principle for a suitable P-function, combined with some geometric arguments involving the mean curvature of ∂Ω.     

On the instability in non-abelian gauge theories with constant magnetic flux

We discuss the stability of Yang-Mills fields in the framework of a sourceless scheme introduced by 't Hooft. We find that abelian magnetic field configurations in Yang-Mills theories are always unstable. A classification for the unstable modes is given.     

Harmonic generation in an ionized gas medium with a 100-femtosecond, high repetition rate laser source at intermediate intensities

 We report the realization of a vacuum-ultraviolet radiation source based on high-order harmonic generation in noble-gas samples, operating at high repetition rate. In particular, we observed up to the 13th harmonic (λ=61 nm) of the fundamental frequency of a short pulse, high repetition rate titanium–sapphire laser after its interaction with a Xe gas jet. The effects of the propagation of the fundamental and harmonic beams through an ionized medium are studied by analysing the spectral profile of the 9th and 7th harmonics. Finally, we report a study of the dependence of the harmonic conversion efficiency on relative position of the focus and the gas target. Received: 29 March 1996/Revised version: 25 July 1996     

Mouse brain expression patterns of <Emphasis Type="Italic">Spg7</Emphasis>, <Emphasis Type="Italic">Afg3l1</Emphasis>, and <Emphasis Type="Italic">Afg3l2</Emphasis> transcripts,encoding for the mitochondrial <Emphasis Type="Italic">m</Emphasis>-AAA protease

Background  

The m-AAA (ATPases Associated with a variety of cellular Activities) is an evolutionary conserved metalloprotease complex located in the internal mitochondrial membrane. In the mouse, it is a hetero-oligomer variably formed by the Spg7, Afg3l1, and Afg3l2 encoded proteins, or a homo-oligomer formed by either Afg3l1 or Afg3l2. In humans, AFG3L2 and SPG7 genes are conserved, whereas AFG3L1 became a pseudogene. Both AFG3L2 and SPG7 are involved in a neurodegenerative disease, namely the autosomal dominant spinocerebellar ataxia SCA28 and a recessive form of spastic paraplegia, respectively.