Existence and multiplicity of solutions for elliptic equations with jumping nonlinearities

In this paper we are concerned with a semilinear elliptic Dirichlet problem with jumping nonlinearity and, using a completely variational method, we show that the number of solutions may be arbitrarily large provided the number of jumped eigenvalues is large enough. In order to prove this fact, we show that for every positive integer k, when a suitable parameter is large enough, there exists a solution which presents k peaks. Under the assumptions we consider in this paper, new (unexpected) phenomena are observed in the study of this problem and new methods are required to construct the k-peaks solutions and describe their asymptotic behavior (weak limits of the rescaled solutions, localization of the concentration points of the peaks, asymptotic profile of the rescaled peaks, etc.).     

Existence of 3D-Vortices in Abelian Gauge Theories

We consider the Abelian gauge theory related to the interaction between the solitary waves of the nonlinear Klein-Gordon equation and the electromagnetic field. We analyze the existence of electromagnetostatic solutions and, in particular, of vortices. Dedicated to the memory of Professor Aldo Cossu     

Periodic unfolding and Robin problems in perforated domains

The periodic unfolding method was introduced in 2002 by D. Cioranescu et al. for the study of classical periodic homogenization. In this Note, we extend this method to perforated domains introducing also a boundary unfolding operator. As an application, we study the homogenization of some elliptic problems with Robin condition on the boundary of the holes. To cite this article: D. Cioranescu et al., C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Multistep Methods for Conservative Problems

We discuss the use of linear multistep methods for the solution of conservative (in particular Hamiltonian) problems. Despite the lack of good results concerning their behaviour, linear multistep methods have been satisfactorily used by researchers in applicative fields. We try to elucidate the requirements that a linear multistep method should fulfill in order to be suitable for the integration of such problems, relating our analysis to their use as boundary value methods. We collect a number of results obtained for the linear autonomous case and also consider some numerical tests on the Kepler problem.Work supported by GNCS and MIUR.     

Multiplicity of positive and nodal solutions for scalar field equations

In this paper the question of finding infinitely many solutions to the problem −Δu+a(x)u=|u|^p−2u$-\mathrm{\Delta }u+a(x)u={|u|}^{p-2}u$, in R^N${\mathbb{R}}^N$, u∈H¹(R^N)$u\in H^1({\mathbb{R}}^N)$, is considered when N≥2$N\ge 2$, p∈(2,2N/(N−2))$p\in (2,2N/(N-2))$, and the potential a(x)$a(x)$ is a positive function which is not required to enjoy symmetry properties. Assuming that a(x)$a(x)$ satisfies a suitable “slow decay at infinity” condition and, moreover, that its graph has some “dips”, we prove that the problem admits either infinitely many nodal solutions or infinitely many constant sign solutions. The proof method is purely variational and allows to describe the shape of the solutions.     

Interactions in water-alcohol mixtures: Conductance of lithium picrate in solutions of water and isomeric pentanols at 25°C

Conductance measurements of lithium picrate in solutions of water in n-pentanol, 2-pentanol, 3-pentanol, 2-methyl-1-butanol and 3-methyl-1-butanol have been carried out at 25°C. Ionic association and conductance were found to change with water content and with the molecular structure of the alcohols (i.e. position of the OH group and degree of branching of the alkyl chain). These results indicate that both conductance and ion pair formation are not the consequence of the simple motion of ions in the electrical field as required by the continuum model. A more realistic approach, involving the internal structure of the solvent mixtures, has been considered.     

Spin–rotation couplings: spinning test particles and Dirac field

The hypothesis of coupling between spin and rotation introduced long ago by Mashhoon is examined in the context of “1 + 3” and “3 + 1” space-time splitting techniques, either in special or in general relativity. Its content is discussed in terms of classical (Mathisson–Papapetrou–Dixon–Souriou model) as well as quantum physics (Foldy–Wouthuysen transformation for the Dirac field in an external field), reviewing and discussing all the relevant theoretical literature concerning the existence of such effect. Some original contributions are also included. Dedicated to Bahram Mashhoon for his 60th birthday.     

Isolation,Structure Determination and Synthesis of New Acetylenic Steroids from the Sponge Calyx nicaaensis

Two acetylenic steroids, cholest-5-en-23-yn-3β-ol ( 5 ) and 26,27-dinorcholest-5-en-23-yn-3β-ol ( 3 ), and another unsaturated steroidalcohol, stigmasta-5,23-dien-3β-ol ( 7 ), were isolated from the sponge Calyx nicaaensis. The structures of these two acetylenic steroids were established by synthesis. Several attempts to synthesize the marine steroids alcohol calysterol ( 1 ), with a cyclopropene-containing side chain, starting from cholest-5-en-23-yn-3β-ol are also recorded. Addition of ethyl-diazo-acetate to the triple bond was performed, but the reduction to the methyl derivative yielded decomposition products.