Positive Least Energy Solutions and Phase Separation for Coupled Schrodinger Equations with Critical Exponent: Higher Dimensional Case

We study the following nonlinear Schr\"{o}dinger system which is related to Bose-Einstein condensate: {displaymath} {cases}-\Delta u +\la_1 u = \mu_1 u^{2^\ast-1}+\beta u^{\frac{2^\ast}{2}-1}v^{\frac{2^\ast}{2}}, \quad x\in \Omega, -\Delta v +\la_2 v =\mu_2 v^{2^\ast-1}+\beta v^{\frac{2^\ast}{2}-1} u^{\frac{2^\ast}{2}}, \quad x\in \om, u\ge 0, v\ge 0 \,\,\hbox{in \om},\quad u=v=0 \,\,\hbox{on \partial\om}.{cases}{displaymath} Here \om\subset \R^N is a smooth bounded domain, $2^\ast:=\frac{2N}{N-2}$ is the Sobolev critical exponent, -\la_1(\om)0 and $\beta\neq 0$, where \lambda_1(\om) is the first eigenvalue of $-\Delta$ with the Dirichlet boundary condition. When \bb=0, this is just the well-known Brezis-Nirenberg problem. The special case N=4 was studied by the authors in (Arch. Ration. Mech. Anal. 205: 515-551, 2012). In this paper we consider {\it the higher dimensional case $N\ge 5$}. It is interesting that we can prove the existence of a positive least energy solution (u_\bb, v_\bb) {\it for any $\beta\neq 0$} (which can not hold in the special case N=4). We also study the limit behavior of (u_\bb, v_\bb) as $\beta\to -\infty$ and phase separation is expected. In particular, u_\bb-v_\bb will converge to {\it sign-changing solutions} of the Brezis-Nirenberg problem, provided $N\ge 6$. In case \la_1=\la_2, the classification of the least energy solutions is also studied. It turns out that some quite different phenomena appear comparing to the special case N=4.Comment: 48 pages. This is a revised version of arXiv:1209.2522v1 [math.AP

Instant preheating mechanism and UHECR

Top-down models assume that the still unexplained Ultra High Energy Cosmic Rays (UHECR's) are the decay products of superheavy particles. Such particles may have been produced by one of the post-inflationary reheating mechanisms and may account for a fraction of the cold dark matter. In this paper, we assess the phenomenological applicability of the simplest instant preheating framework not to describe a reheating process, but as a mechanism to generate relic supermassive particles as possible sources of UHECR's. We use cosmic ray flux and cold dark matter observational data to constrain the parameters of the model.Comment: 7 pages, 2 figures, submitted to PR

Vacuum local and global electromagnetic self-energies for a point-like and an extended field source

We consider the electric and magnetic energy densities (or equivalently field fluctuations) in the space around a point-like field source in its ground state, after having subtracted the spatially uniform zero-point energy terms, and discuss the problem of their singular behavior at the source's position. We show that the assumption of a point-like source leads, for a simple Hamiltonian model of the interaction of the source with the electromagnetic radiation field, to a divergence of the renormalized electric and magnetic energy density at the position of the source. We analyze in detail the mathematical structure of such singularity in terms of a delta function and its derivatives. We also show that an appropriate consideration of these singular terms solves an apparent inconsistency between the total field energy and the space integral of its density. Thus the finite field energy stored in these singular terms gives an important contribution to the self-energy of the source. We then consider the case of an extended source, smeared out over a finite volume and described by an appropriate form factor. We show that in this case all divergences in local quantities such as the electric and the magnetic energy density, as well as any inconsistency between global and space-integrated local self-energies, disappear.Comment: 8 pages. The final publication is available at link.springer.co

Tetracycline: production, waste treatment and environmental impact assessment

The frequent occurrence of pharmaceuticals in the aquatic environment requires an assessment of their environmental impact and their negative effects in humans. Among the drugs with high harmful potential to the environment are the antibiotics that reach the environment not only, as may be expected, through the effluents from chemical and pharmaceutical industries, but mainly through the sewage and livestock; because around 25 to 75% of the ingested drugs are excreted in unchanged form after the passage through the Gastro-Intestinal Tract. Tetracycline has high world consumption, representing a human consumption of about 23 kg/day in Brazil in 2007. At the moment, researches are being made to develop new tetracycline that incorporate heavy metals (Hg, Cd, Re, Pt, Pd) to their structures in order to increase their bactericidal effect. The conventional wastewater treatment plants are not able to degrade complex organic molecules to reduce their toxicity and improve their biodegradability. For this reason new technologies, i.e., the advanced oxidation processes, are being developed to handle this demand. The objectives of this study are to review the literature on the processes of obtaining tetracycline, presenting its waste treatment methods and evaluation of their environmental impact