Uniqueness and existence of solutions for a singular system with nonlocal operator via perturbation method

In this work, we investigate the existence and the uniqueness of solutions for the nonlocal elliptic system involving a singular nonlinearity as follows: $$ \left\{\begin{array}{ll} (-\Delta_p)^su = a(x)|u|^{q-2}u +\frac{1-\alpha}{2-\alpha-\beta} c(x)|u|^{-\alpha}|v|^{1-\beta}, \quad \text{in }\Omega,\ (-\Delta_p)^s v= b(x)|v|^{q-2}v +\frac{1-\beta}{2-\alpha-\beta} c(x)|u|^{1-\alpha}|v|^{-\beta}, \quad \text{in }\Omega,\ u=v = 0 ,\;\;\mbox{ in }\,\mathbb{R}^N\setminus\Omega, \end{array} \right. $$ where $\Omega $ is a bounded domain in $\mathbb{R}^{n}$ with smooth boundary, $0<\alpha <1,$ $0<\beta <1,$ $2-\alpha -\beta 相似文献   

Size effect in the spin glass magnetization of thin AuFe films as studied by polarized neutron reflectometry

We used polarized neutron reflectometry to determine the temperature dependence of the magnetization of thin AuFe films with 3% Fe concentration. We performed the measurements in a large magnetic field of 6 T in a temperature range from 295 to 2 K. For the films in the thickness range from 500 to 20 nm we observed a Brillouin-type behavior from 295 K down to 50 K and a constant magnetization of about 0.9 micro(B) per Fe atom below 30 K. However, for the 10 nm thick film we observed a Brillouin-type behavior down to 20 K and a constant magnetization of about 1.3 micro(B) per Fe atom below 20 K. These experiments are the first to show a finite-size effect in the magnetization of single spin-glass films in large magnetic fields. Furthermore, the ability to measure the deviation from the paramagnetic behavior enables us to prove the existence of the spin-glass state where other methods relying on a cusp-type behavior fail.     

Multiplicity of positive solutions for a singular and critical problem

In this paper, we study the following singular, critical elliptic problem :

     

W0^1p（x） Versus C^1 Local Minimizers for a Functional with Critical Growth

Let Ω IR^N, （N ≥ 2） be a bounded smooth domain, p is Holder continuous on Ω^-,
1 〈 p^- ：= inf pΩ（x） ≤ p＋ = supp（x） Ω〈∞,
and f：Ω^-× IR be a C^1 function with f（x,s） ≥ 0, V （x,s） ∈Ω × R^＋ and sup ∈Ωf（x,s） ≤ C（1＋s）^q（x）, Vs∈IR^＋,Vx∈Ω for some 0〈q（x） ∈C（Ω^-） satisfying 1 〈p（x） 〈q（x） ≤p^＊ （x） -1, Vx ∈Ω ^- and 1 〈 p^- ≤ p^＋ ≤ q- ≤ q＋. As usual, p＊ （x） = Np（x）/N-p（x） if p（x） 〈 N and p^＊ （x） = ∞- if p（x） if p（x） 〉 N. Consider the functional I： W0^1,p（x） （Ω） →IR defined as
I（u） def= ∫Ω1/p（x）｜△｜^p（x）dx-∫ΩF（x,u^＋）dx,Vu∈W0^1,p（x）（Ω）,
where F （x, u） = ∫0^s f （x,s） ds. Theorem 1.1 proves that if u0 ∈ C^1 （Ω^-） is a local minimum of I in the C1 （Ω^-） ∩C0 （Ω^-）） topology, then it is also a local minimum in W0^1,p（x） （Ω）） topology. This result is useful for proving multiple solutions to the associated Euler-lagrange equation （P） defined below.     

Theoretical Study of Structural,Electronic, Optical,and Elastic Properties of KLiX (X: S,Se, and Te) under Hydrostatic Pressure: A Pseudo Potential Plane Wave (PP-PW) Contribution

Crystallography Reports - We have studied structural, electronic, optical, and elastic properties of KLiX under pressure using the Density Functional Theory (DFT) within the Generalized Gradient...     

Littlewood–Paley decomposition in quantum calculus

ABSTRACT

We introduce Littlewood–Paley decomposition related to q-Rubin's operator, this allows us to provide a dyadic characterization of Sobolev, Hölder and Lebesgue spaces associated with the q-Rubin's operators and to establish some embedding results for these spaces. We construct the paraproduct operators associated with the q-Rubin's operators and we establish its action on the Sobolev and Hölder spaces.     

Purification and Biochemical Characterization of a Highly Thermostable Xylanase from <Emphasis Type="Italic">Actinomadura</Emphasis> sp. Strain Cpt20 Isolated from Poultry Compost

An extracellular thermostable xylanase from a newly isolated thermophilic Actinomadura sp. strain Cpt20 was purified and characterized. Based on matrix-assisted laser desorption–ionization time-of-flight mass spectrometry analysis, the purified enzyme is a monomer with a molecular mass of 20,110.13 Da. The 19 residue N-terminal sequence of the enzyme showed 84% homology with those of actinomycete endoxylanases. The optimum pH and temperature values for xylanase activity were pH 10 and 80 °C, respectively. This xylanase was stable within a pH range of 5–10 and up to a temperature of 90 °C. It showed high thermostability at 60 °C for 5 days and half-life times at 90 °C and 100 °C were 2 and 1 h, respectively. The xylanase was specific for xylans, showing higher specific activity on soluble oat-spelt xylan followed by beechwood xylan. This enzyme obeyed the Michaelis–Menten kinetics, with the K _m and k _cat values being 1.55 mg soluble oat-spelt xylan/ml and 388 min⁻¹, respectively. While the xylanase from Actinomadura sp. Cpt20 was activated by Mn²⁺, Ca²⁺, and Cu²⁺, it was, strongly inhibited by Hg²⁺, Zn²⁺, and Ba²⁺. These properties make this enzyme a potential candidate for future use in biotechnological applications particularly in the pulp and paper industry.     

Influence of Mn doping on structural and optical properties of ZnO nano thin films synthesized by sol–gel technique

Undoped and Mn-doped ZnO samples with different percentages of Mn content (1, 5 and 10?at%) were synthesized by a dip-coating sol?Cgel method. We have studied the structural, chemical and optical properties of the samples by using X-ray diffraction (XRD), scanning electron microscopy (SEM) and UV-visible spectroscopy. The XRD spectra show that all the samples are hexagonal wurtzite structures. We note that doping favors c-axis orientation along (002) planes. Up to 5?at% of Mn doping level, the c-axis lattice parameter shifts towards higher values with the increase of manganese content in the films. The expansion of the lattice constant of ZnO?CMn indicates that Mn is really doped into the ZnO. The SEM investigations of all samples revealed that the crystallites are of nanometer size. The surface quality of the ZnO?CMn film increases with Mn doping but no significant change of the grain size is observed from SEM images. The transmittance spectra show that the transparency of all the samples is greater than 85?%. We note, also, that a small doping (1?%) lowered the refractive index while the thickness of the layers and the gap increase. However, on raising the proportion of Mn beyond 5?%, practically the same values of index and gap as pure ZnO are found.     

Optical measurements of ZnS nanoparticles aqueous solution

We synthesized zinc sulfide (ZnS) nanopowders with size ranging from 2 to 100 nm by a simple, low-cost, and mass production chemical method. The nanoparticles (NPs) were characterized by X-ray powder diffraction (XRD), atomic force microscopy (AFM), transmission electron microscopy (TEM), and UV-vis absorption spectroscopy. Our study concerns also the change in the refractive index of deionized water in presence of ZnS nanospheres. We present experimental results on effective index variation of water dispersed ZnS NPs at different wavelengths in visible spectrum.