Two solutions for a singular elliptic equation with critical growth at infinity

We look for positive solutions for the singular equation $-\Delta u-\frac{1}{2}\left(x\cdot \nabla u\right)=\mu h\left(x\right)u^{q-1}+\lambda u+u^{(N+2)/(N-2)},\text{in }{\mathbb{R}}^N,$ where $N\ge 3$, $\lambda >0$, $\mu >0$ is a parameter, $0<q<1$ and $h$ has some summability properties. By using a perturbation method and critical point theory, we obtain two solutions when $max\{1,N/4\}<\lambda <N/2$ and the parameter $\mu >0$ is small.     

Template‐Engaged Solid‐State Synthesis of Barium Magnesium Silicate Yolk@Shell Particles and Their High Photoluminescence Efficiency

This study presents a new synthetic method for fabricating yolk@shell‐structured barium magnesium silicate (BMS) particles through a template‐engaged solid‐state reaction. First, as the core template, (BaMg)CO₃ spherical particles were prepared based on the coprecipitation of Ba²⁺ and Mg²⁺. These core particles were then uniformly shelled with silica (SiO₂) by using CTAB as the structure‐directing template to form (BaMg)CO₃@SiO₂ particles with a core@shell structure. The (BaMg)CO₃@SiO₂ particles were then converted to yolk@shell barium magnesium silicate (BMS) particles by an interfacial solid‐state reaction between the (BaMg)CO₃ (core) and the SiO₂ (shell) at 750 °C. During this interfacial solid‐state reaction, Kirkendall diffusion contributed to the formation of yolk@shell BMS particles. Thus, the synthetic temperature for the (BaMg)SiO₄:Eu³⁺ phosphor is significantly reduced from 1200 °C with the conventional method to 750 °C with the proposed method. In addition, the photoluminescence intensity of the yolk@shell (BaMg)SiO₄:Eu³⁺phosphor was found to be 9.8 times higher than that of the conventional (BaMg)SiO₄:Eu³⁺ phosphor. The higher absorption of excitation light by the structure of the yolk@shell phosphor is induced by multiple light‐reflection and ‐scattering events in the interstitial void between the yolk and the shell. When preparing the yolk@shell (BaMg)SiO₄:Eu³⁺ phosphor, a hydrogen environment for the solid‐state reaction results in higher photoluminescence efficiency than nitrogen and air environments. The proposed synthetic method can be easily extended to the synthesis of other yolk@shell multicomponent metal silicates.     

On a discretization process of fractional-order Logistic differential equation

In this work we introduce a discretization process to discretize fractional-order differential equations. First of all, we consider the fractional-order Logistic differential equation then, we consider the corresponding fractional-order Logistic differential equation with piecewise constant arguments and we apply the proposed discretization on it. The stability of the fixed points of the resultant dynamical system and the Lyapunov exponent are investigated. Finally, we study some dynamic behavior of the resultant systems such as bifurcation and chaos.     

Oscillations of nonlinear hyperbolic equations with functional arguments via Riccati method

This paper is devoted to the study of the oscillatory behavior of solutions of nonlinear hyperbolic equations with functional arguments by using integral averaging method and a generalized Riccati technique. First, we establish oscillation results for nonlinear hyperbolic equations by applying oscillation criteria for Riccati inequality. Secondly, we present interval oscillation results for nonlinear hyperbolic equations. These results improve and extend oscillation results of Cui and Xu [1].     

Exact controllability and continuous dependence of fractional neutral integro-differential equations with state-dependent delay

In the present paper, with the help of the resolvent operator and some analytic methods, the exact controllability and continuous dependence are investigated for a fractional neutral integro-differential equations with state-dependent delay. As an application, we also give one example to demonstrate our results.     

Intrinsic ultracontractivity on Riemannian manifolds with infinite volume measures

By establishing the intrinsic super-Poincar'e inequality,some explicit conditions are presented for diffusion semigroups on a non-compact complete Riemannian manifold to be intrinsically ultracontractive.These conditions,as well as the resulting uniform upper bounds on the intrinsic heat kernels,are sharp for some concrete examples.