Eigenvalue problems with weights in Lorentz spaces

Given V, w locally integrable functions on a general domain Ω with V ≥  0 but w allowed to change sign, we study the existence of ground states for the nonlinear eigenvalue problem:

$-\Delta u + V u = \lambda w |u|^{p-2} u, \quad u|_{\partial \Omega} =0,$

with p subcritical. These are minimizers of the associated Rayleigh quotient whose existence is ensured under suitable assumptions on the weight w. In the present paper we show that an admissible space of weight functions is provided by the closure of smooth functions with compact support in the Lorentz space \({L(\tilde p,\infty)}\) with \({\frac{1}{{\widetilde p}} + \frac{p}{2^{\star}} =1}\) . This generalizes previous results and gives new sufficient conditions ensuring existence of extremals for generalized Hardy–Sobolev inequalities. The existence in such a generality of a principal eigenfunction in the linear case p = 2 is applied to study the bifurcation for semilinear problems of the type

$-\Delta u= \lambda (a(x)u + b(x) r(u)),$

where a, b are indefinite weights belonging to some Lorentz spaces, and the function r has subcritical growth at infinity.

     

A blowing-up branch of solutions for a mean field equation

We consider the equation

Solid-Phase Molecular Recognition of Cytosine Based on Proton-Transfer Reaction

Abstract  The proton-transfer molecular adducts cytosine/isoorotic acid (1:1) dihydrate (CytIsor) and cytosine/maleic hydrazide (2:2) dihydrate (CytMal) have been studied by X-ray diffraction methods. CytIsor crystallizes in the orthorhombic system, space group Pbnb, a = 7.4859 (4), b = 12.7977 (9), c = 26.4573 (16) ?, V = 2534.7 (3) ?³. CytMal crystallizes in the triclinic system, space group P − 1, a = 6.7767 (9), b = 12.063 (2), c = 12.937 (3) ?, α = 78.58 (2), β = 75.87 (2), γ = 75.63 (1)°, V = 982.8 (3) ?³. In both compounds protonation occurs at the N3 atom of the cytosine aminooxo tautomer as a result of the proton-transfer process from the acidic hydroxy groups of the counterions. In the crystal structure, both cocrystals are stabilized by a plethora of N–H···O, N–H···N and O–H···O hydrogen bonds. It is shown the ability of cytosine to form large organic assemblies, with the appropriate guest, due to solid-state recognition involving synthons based on hydrogen bonding between functional groups as sites of proton-transfer reactions. Index Abstract  The proton-transfer molecular adducts of cytosine with isoorotic acid and with maleic hydrazide have been synthesized in the solid state and their hydrogen-bonding patterns have been studied by X-ray diffraction methods.      

Numerical solution of the Helmholtz equation in an infinite strip by Wiener‐Hopf factorization

We describe an algorithm to compute numerically the solution of the Helmholtz equation: Δu + κu = f, u ∈ H₀¹(S), where S is an infinite strip and κ a given bounded function. By using the finite difference approximation on the entire strip, we are led to solve an infinite linear system. When κ is constant the associated matrix is block Toeplitz and banded and the system can be solved using a Wiener‐Hopf factorization. This approach can also be adapted to deal with the case when κ is constant outside a bounded domain of the strip. Numerical results are given to assess the performance of our method. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

Finite Anticlastic Bending of Hyperelastic Solids and Beams

This paper deals with the equilibrium problem in nonlinear elasticity of hyperelastic solids under anticlastic bending. A three-dimensional kinematic model, where the longitudinal bending is accompanied by the transversal deformation of cross sections, is formulated. Following a semi-inverse approach, the displacement field prescribed by the above kinematic model contains three unknown parameters. A Lagrangian analysis is performed and the compressible Mooney-Rivlin law is assumed for the stored energy function. Once evaluated the Piola-Kirchhoff stresses, the free parameters of the kinematic model are determined by using the equilibrium equations and the boundary conditions. An Eulerian analysis is then accomplished to evaluating stretches and stresses in the deformed configuration. Cauchy stress distributions are investigated and it is shown how, for wide ranges of constitutive parameters, the obtained solution is quite accurate. The whole formulation proposed for the finite anticlastic bending of hyperelastic solids is linearized by introducing the hypothesis of smallness of the displacement and strain fields. With this linearization procedure, the classical solution for the infinitesimal bending of beams is fully recovered.     

Metal Complex Formation and Anticancer Activity of Cu(I) and Cu(II) Complexes with Metformin

Metformin has been used for decades in millions of type 2 diabetes mellitus patients. In this time, correlations between metformin use and the occurrence of other disorders have been noted, as well as unpredictable metformin side effects. Diabetes is a significant cancer risk factor, but unexpectedly, metformin-treated diabetic patients have lower cancer incidence. Here, we show that metformin forms stable complexes with copper (II) ions. Both copper(I)/metformin and copper(II)/metformin complexes form adducts with glutathione, the main intracellular antioxidative peptide, found at high levels in cancer cells. Metformin reduces cell number and viability in SW1222 and K562 cells, as well as in K562-200 multidrug-resistant cells. Notably, the antiproliferative effect of metformin is enhanced in the presence of copper ions.     

Two-pion production in proton-proton collisions with a polarized beam

The two-pion production reaction pp → ppπ⁺π⁻ was measured with a polarized proton beam at T _p ≈ 750 and 800MeV using the short version of the COSY-TOF spectrometer. The implementation of a delayed-pulse technique for Quirl and central calorimeter provided positive π⁺ identification in addition to the standard particle identification, energy determination as well as time-of-flight and angle measurements. Thus all four-momenta of the emerging particles could be determined with 1–4 overconstraints. Total and differential cross-sections as well as angular distributions of the vector analyzing power have been obtained. They are compared to previous data and theoretical calculations. In contrast to predictions we find significant analyzing-power values up to A _y = 0.3. The data taken in the energy region of the excitation of the Roper resonance confirm that its dominant ππ decay channel is N ^* → Nσ.     

Conformational studies by nuclear magnetic resonance in nematic phases: Benzyl chloride

Spectra of benzyl chloride were analysed in two nematic phases (PCH and ZLI). The existence of the molecule as conformer with the C? Cl bond perpendicular or parallel to the benzene plane can be safely excluded. The data are consistent with a conformer with the C? Cl bond forming an angle of 60° with the benzene plane. A full discussion of the conformational problem was not possible owing to the small number of independent D_ij couplings related to the internal motion. In addition, when dealing with low potential barriers, the LCNMR technique does not give rigorous answers.     

On the Number of Rational Points of Bounded Height on Smooth Bilinear Hypersurfaces in Biprojective Space

Asymptotic formulae for the number of rational points of boundedheight on flag varieties have earlier been established. In thepaper these asymptotic formulae are recovered by a new methodfor varieties in biprojective space defined over Q that areisomorphic to the flag variety of lines in hyperplanes. The result is obtained by an application of Heath-Brown's newform of the circle method. It serves as a pointer to the investigationof rational points of bounded height on varieties in multiprojectivespace.     

A Neumann problem in exterior domain

We are concerned with the existence of radial solutions for the following Neumann problem