Nodal Multi-peak Standing Waves of Fourth-Order Schrödinger Equations with Mixed Dispersion

We consider the existence and concentration properties of standing waves for a fourth-order Schrödinger equation with mixed dispersion, which was introduced to regularize and stabilize solutions to the classical time-dependent Schrödinger equation. This leads to study multi-peak solutions to the following singularly perturbed fourth-order nonlinear Schrödinger equation $$\begin{aligned} {\varepsilon ^{\text {4}}}{\Delta ^{\text {2}}}u - \beta {\varepsilon ^2}\Delta u + V(x)u = |u{|^{p - 2}}u{\text { in }}{\mathbb {R}^N},{\text { }}u \in {H^2}({\mathbb {R}^N}). \end{aligned}$$ ε 4 Δ 2 u - β ε 2 Δ u + V ( x ) u = | u | p - 2 u in R N , u ∈ H 2 ( R N ) . We first establish a local $${W^{4,p}}$$ W 4 , p -estimate for a class of fourth-order semilinear elliptic equations, which is a key to get the uniform and global $${L^\infty }$$ L ∞ -estimate of solutions to the considered singularly perturbed equation above. Next, under certain assumptions on $$\beta $$ β and the potential V(x), we construct a family of sign-changing multi-peak solutions with a unique maximum (or minimum) point on each component. We prove that these solutions concentrate around any prescribed finite set of local minima (possibly degenerate) of the potential V(x). Compared with the classical singularly perturbed Schrödinger equation, the presence of a fourth-order term in the problem above forces the development of new techniques to obtain qualitative properties of multi-peak solutions.

ENERGY CONCENTRATION PROPERTIES OF A p-GINZBURG–LANDAU MODEL

This paper is concerned with the p-Ginzburg–Landau (p-GL) type model with $p\neq 2$ . First, we obtain global energy estimates and energy concentration properties by the singularity analysis. Next, we give a decay rate of $1-|u_\varepsilon |$ in the domain away from the singularities when $\varepsilon \to 0$ , where $u_\varepsilon $ is a minimizer of p-GL functional with $p \in (1,2)$ . Finally, we obtain a Liouville theorem for the finite energy solutions of the p-GL equation on $\mathbb {R}^2$ .

Multiplicity of concentrating solutions for a class of magnetic Schrödinger-Poisson type equation

In this paper, we study the following nonlinear magnetic Schrödinger-Poisson type equation $$\begin{array}{} \displaystyle \left\{\!\begin{aligned}&\Big(\frac{\varepsilon}{i}\nabla-A(x)\Big)^{2}u+V(x)u+\epsilon^{-2}(\vert x\vert^{-1}\ast \vert u\vert^{2})u = f(|u|^{2})u\quad\hbox{in }\mathbb{R}^3,\\&u\in H^{1}(\mathbb{R}^{3}, \mathbb{C}),\end{aligned}\right. \end{array}$$ where ϵ > 0, V : ℝ3 → ℝ and A : ℝ3 → ℝ3 are continuous potentials. Under a local assumption on the potential V, by variational methods, penalization technique, and Ljusternick-Schnirelmann theory, we prove multiplicity and concentration properties of nontrivial solutions for ε > 0 small. In this problem, the function f is only continuous, which allow to consider larger classes of nonlinearities in the reaction.

Use of Raw and Physically Modified Rice Starches as Fat Replacer in Whipping Cream

Raw, retrograded and retrograded-annealed starches obtained in a previous novel study from three rice varieties widely differing in apparent amylose content (22.7%, 9.8% and 0.3%) were applied for partially replacing fat in fresh cream to prepare to the consistency of whipping cream with approximately 15% final fat concentration. Properties of the whipped creams were studied and compared with a commercial whipping cream taken as standard. Differences between the mean values were assessed by Duncan’s multiple range tests at a significance level of 95%. Fat replacement resulted in whipping time as low as 60 seconds and improved foam stability of the whipped creams with significant overrun (up to 44%), suggesting industrial applicability of the starch samples as fat replacers. Incorporated starch resulted in better water retention and structural stability lower weeping out of liquid upon freezing and thawing. Modified waxy starch substitution resulted in cream texture closest to the commercial cream standard, suggesting efficient fat replacement. Thereby, starch incorporated whipping cream with more than 62% lower fat content than commercial variants could be obtained. This would thereby help in lower glycemic index, low calorie and lower priced alternative to common fat-rich whipping creams.

Explosion Pressure and Minimum Explosible Concentration Properties of Metal Sulfide Ore Dust Clouds

The explosion pressure and minimum explosible concentration (MEC) properties of metal sulfide ore dust clouds are valuable for the prevention and control of metal sulfide ore dust explosions. In this study, a 20 L explosion sphere vessel was used to investigate the effect of sulfur content, particle size, and concentration on the explosion pressure and minimum explosible concentration of metal sulfide ore dust clouds. Four samples with different sulfur contents were selected (30%–40%, 20%–30%, 10%–20%, and 0%–10%). Before and after the explosion, samples were tested by X-ray diffraction. The results indicate that the metal sulfide ore dust is explosive dust with St1 grade explosion pressure. With an increase in concentration, the maximum explosion pressure increased at first and then decreased. With an increase in sulfide content, the explosion pressure of metal sulfide ore dust increased, while the minimum explosible concentration decreased. As particle size decreased, the MEC also decreased. The sulfur content, particle size, and concentration of metal sulfide ore dust were the main factors affecting the explosion hazard.

Properties of PDMS-divinylbenzene based pre-concentrators for nitroaromatic vapors

The mode of DVB incorporation into PDMS strongly influences the structure and pre-concentration properties of the resulting film.