A note on fractional $${\varvec{}}$$-Lalacian roblems with singular weights

We study a class of fractional p-Laplacian problems with weights which are possibly singular on the boundary of the domain. We provide existence and multiplicity results as well as characterizations of critical groups and related applications.     

Solution‐Based Synthesis of Layered Intergrowth Compounds of the Homologous PbmBi2nTe3n+m Series as Nanosheets

Layered intergrowth compounds in the homologous Pb_mBi_2nTe_3n+m family are interesting because they are examples of natural heterostructures. We present a simple solution‐based synthesis of two‐dimensional nanosheets of PbBi₂Te₄, Pb₂Bi₂Te₅, and PbBi₆Te₁₀ layered intergrowth compounds, which are members of the Pb_mBi_2nTe_3n+m [that is, (PbTe)_m(Bi₂Te₃)_n] homologous series. Few‐layer nanosheets exhibit narrow optical band gaps (0.25–0.7 eV) with semiconducting electronic‐transport properties.     

An Indium‐Seamed Hexameric Metal–Organic Cage as an Example of a Hexameric Pyrogallol[4]arene Capsule Conjoined Exclusively by Trivalent Metal Ions

A hexameric metal–organic nanocapsule is assembled from pyrogallol[4]arene units, which are stitched together with indium ions. This indium‐seamed capsule is the first instance of a M₂₄L₆ type hexameric coordination cage held together exclusively by trivalent metal ions. Explicitly, unlike previously reported pyrogallol[4]arene‐based metal‐seamed capsules, the current In³⁺ seamed capsule is entirely supported by O→In coordinate bonds. This work demonstrates the important proof of concept of the ability of pyrogallol[4]arene to react with metals in higher oxidation states to assemble into atomically‐precise hexameric coordination cages. As such, these results open up exciting avenues toward the assembly of previously unanticipated metal–organic capsules, for example offering inspiration for tackling metals exhibiting high valence states such as in the lanthanide and actinide series.     

The Fucik spectrum and critical groups

We compute critical groups of zero for variational functionals arising from semilinear elliptic boundary value problems with jumping nonlinearities when the asymptotic limits of the nonlinearity fall in certain parts of Type (II) regions between curves of the Fucik spectrum.

     

An existence result for a class of quasilinear elliptic eigenvalue problems in unbounded domains

We consider a nonlinear eigenvalue problem under Robin boundary conditions in a domain with (possibly noncompact) smooth boundary. The problem involves a weighted p–Laplacian operator and subcritical nonlinearities satisfying Ambrosetti–Rabinowitz type conditions. Using Morse theory and a cohomological local splitting as in Degiovanni et al. (Commun Contemp Math 12:475–486, 2010), we prove the existence of a nontrivial weak solution for all (real) values of the eigenvalue parameter. Our result is new even in the semilinear case p = 2 and complements some recent results obtained in Autuori et al. (Adv Anal Equ 18:1–48, 2013).     

Existence of multiple positive solutions for second order nonlinear dynamic BVPs by variational methods

The aim of this paper is to employ variational techniques and critical point theory to prove some sufficient conditions for the existence of multiple positive solutions to a nonlinear second order dynamic equation with homogeneous Dirichlet boundary conditions.     

Synthesis in ionic liquids: [Bi2Te2Br](AlCl4), a direct gap semiconductor with a cationic framework

The Lewis acidic ionic liquid EMIMBr-AlCl(3) (EMIM = 1-ethyl-3-methylimidazolium) allows a novel synthetic route to the semiconducting layered metal chalcogenides halide [Bi(2)Te(2)Br](AlCl(4)) and its Sb analogue. [Bi(2)Te(2)Br](AlCl(4)) is a direct band gap, strongly anisotropic semiconductor and consists of cationic infinite layers of [Bi(2)Te(2)Br](+) and [AlCl(4)](-) anions inserted between the layers.     

Soft Phonon Modes Leading to Ultralow Thermal Conductivity and High Thermoelectric Performance in AgCuTe

Crystalline solids with intrinsically low lattice thermal conductivity (κ_L) are crucial to realizing high‐performance thermoelectric (TE) materials. Herein, we show an ultralow κ_L of 0.35 Wm^?1 K^?1 in AgCuTe, which has a remarkable TE figure‐of‐merit, zT of 1.6 at 670 K when alloyed with 10 mol % Se. First‐principles DFT calculation reveals several soft phonon modes in its room‐temperature hexagonal phase, which are also evident from low‐temperature heat‐capacity measurement. These phonon modes, dominated by Ag vibrations, soften further with temperature giving a dynamic cation disorder and driving the superionic transition. Intrinsic factors cause an ultralow κ_L in the room‐temperature hexagonal phase, while the dynamic disorder of Ag/Cu cations leads to reduced phonon frequencies and mean free paths in the high‐temperature rocksalt phase. Despite the cation disorder at elevated temperatures, the crystalline conduits of the rigid anion sublattice give a high power factor.     

Use of ionic liquids in the synthesis of nanocrystals and nanorods of semiconducting metal chalcogenides

We have synthesized nanoparticles of hexagonal CdS in the diameter range 3-13 nm by the reaction of cadmium acetate dihydrate with thioacetamide in imidazolium [BMIM]-based ionic liquids. We have obtained three different particle sizes of CdS by changing the anion of the ionic liquid. Addition of trioctylphosphine oxide (TOPO) to the reaction mixture causes greater monodispersity as well as smaller particle size, while addition of ethylenediamine produces nanorods of 7 nm average diameter. Hexagonal ZnS and cubic PbS nanoparticles with average diameters of 3 and 10 nm, respectively, have been prepared by the reaction of the metal acetates with thioacetamide in [BMIM][BF4]. Hexagonal CdSe nanoparticles with an average diameter 12 nm were obtained by the reaction of cadmium acetate dihydrate with dimethylselenourea in [BMIM][BF4]. In this case also we observe the same effect of the addition of TOPO as in the case of CdS. Addition of ethylenediamine to the reaction mixture gives rise to nanorods. ZnSe nanowires with a cubic structures, possible diameters in the range 70-100 nm by the reaction of zinc acetate dihydrate with dimethylselenourea in [BMIM][MeSO4]. The nanostructures obtained are single crystalline in all the cases. Most of the nanostructures show characteristic UV/Vis absorption and photoluminescence emission spectra. The thermodynamically most stable structures are generally produced in the synthesis carried out in ionic liquids.