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.     

Elliptic equations with jumping nonlinearities involving high eigenvalues

We present some multiplicity results concerning semilinear elliptic Dirichlet problems with jumping nonlinearities where the jumping condition involves a set of high eigenvalues not including the first one. Using a variational method we show that the number of solutions may be arbitrarily large provided the number of jumped eigenvalues is large enough. Indeed, we prove that for every positive integer $k$ there exists a positive integer $n(k)$ such that, if the number of jumped eigenvalues is greater than $n(k),$ then the problem has at least a solution which presents $k$ peaks. Moreover, we describe the asymptotic behaviour of the solutions as the number of jumped eigenvalues tends to infinity; in particular, we analyse some concentration phenomena of the peaks (near points or suitable manifolds), we describe the asymptotic profile of the rescaled peaks, etc $\ldots $      

Exclusively Heteronuclear NMR Experiments to Obtain Structural and Dynamic Information on Proteins

Provided that ¹³C‐detected NMR experiments are either preferable or complementary to ¹H detection, we report here tools to determine C^α? C′, C′? N, and C^α? H^α residual dipolar couplings on the basis of the CON experiment. The coupling constants determined on ubiquitin are consistent with the subset measured with the ¹H‐detected HNCO sequences. Since the utilization of residual dipolar couplings may depend on the mobility of the involved nuclei, we also provide tools to measure longitudinal and transverse relaxation rates of N and C′. This new set of experiments is a further development of a whole strategy based on ¹³C direct‐detection NMR spectroscopy for the study of biological macromolecules.     

Numerical transformation methods: a constructive approach

Within group invariance theory we consider a constructive approach in order to define numerical transformation methods. These are initial-value methods for the solution of boundary value problems governed by ordinary differential equations. Here we consider the class of free boundary value problems governed by the most general second-order equation in normal form. For this class of problems the main theorem is concerned with the definition of an iterative transformation method. The definition of a noniterative method, applicable to a subclass of the original class of problems, follows as a corollary. Therefore, the proposed constructive approach allows us to establish a unifying framework for noniterative and iterative transformation methods.     

Hydrogen–hydrogen bonding: The current density perspective

Current density plots of closed‐shell intermolecular H? H interactions characterized by a bond critical point (BCP) show two vortices separated by a saddle, a pattern which allows for a clear definition of a pair current strength. This H? H current strength turns out to be roughly related to the potential energy density at the BCP and then to the dissociation energy. The same pattern is also recognizable, at least for an azimuthal orientation of a field perpendicular to the H? H line, for the intramolecular interactions previously investigated to propose the H? H bonding. In the case of the H atoms of the bay region of polycyclic aromatic hydrocarbons, the current of the H? H delocalized diatropic vortex gives a quantitative indication of stabilization; however, on rotation of the field and the subsequent onset of a bay‐delocalized paratropic vortex (a typical signature of antiaromaticity), the diatropic vortex can be reshaped or it can even disappear, consistently with its smallness, and thus showing the effect of other more relevant interactions. © 2015 Wiley Periodicals, Inc.     

Quasi-periodic water waves

We present the result and the ideas of the recent paper (Berti and Montalto, Quasi-periodic standing wave solutions of gravity-capillary water waves, http://arxiv.org/abs/1602.02411, 2016) concerning the existence of Cantor families of small-amplitude time quasi-periodic standing wave solutions (i.e. periodic and even in the space variable x) of a 2-dimensional ocean, with infinite depth, in irrotational regime, under the action of gravity and surface tension at the free boundary. These quasi-periodic solutions are linearly stable.