Integrability and asymptotics of positive solutions of a γ-Laplace system

AbstractIn this paper, we use the potential analysis to study the properties of the positive solutions of a γ-Laplace system in Rn−div(|∇u|γ−2∇u)=upvq,−div(|∇v|γ−2∇v)=vpuq. Here 1<γ⩽2, p,q>0 satisfy the critical condition p+q=γ⁎−1. First, the positive solutions u and v satisfy an integral system involving the Wolff potentials. We then use the method of regularity lifting to obtain an optimal integrability for this Wolff type integral system. Different from the case of γ=2, it is more difficult to handle the asymptotics since u and v have not radial structures. We overcome this difficulty by a new method and obtain the decay rates of u and v as |x|→∞. We believe that this new method is appropriate to deal with the asymptotics of other decaying solutions without the radial structures

Linear Difference Equations with a Transition Point at the Origin

A pair of linearly independent asymptotic solutions are constructed for the second-order linear difference equation {equation*} P_{n+1}(x)-(A_{n}x+B_{n})P_{n}(x)+P_{n-1}(x)=0, {equation*} where $A_n$ and $B_n$ have asymptotic expansions of the form {equation*} A_n\sim n^{-\theta}\sum_{s=0}^\infty\frac{\alpha_s}{n^s},\qquad B_n\sim\sum_{s=0}^\infty\frac{\beta_s}{n^s}, {equation*} with $\theta\neq0$ and $\alpha_0\neq0$ being real numbers, and $\beta_0=\pm2$. Our result hold uniformly for the scaled variable $t$ in an infinite interval containing the transition point $t_1=0$, where $t=(n+\tau_0)^{-\theta} x$ and $\tau_0$ is a small shift. In particular, it is shown how the Bessel functions $J_\nu$ and $Y_\nu$ get involved in the uniform asymptotic expansions of the solutions to the above three-term recurrence relation. As an illustration of the main result, we derive a uniform asymptotic expansion for the orthogonal polynomials associated with the Laguerre-type weight $x^\alpha\exp(-q_mx^m)$, $x>0$, where $m$ is a positive integer, $\alpha>-1$ and $q_m>0$.Comment: 33 pages, reference update

Simultaneous 3D Construction and Imaging of Plant Cells Using Plasmonic Nanoprobe Assisted Multimodal Nonlinear Optical Microscopy

Nonlinear optical (NLO) imaging has emerged as a promising plant cell imaging technique due to its large optical penetration, inherent 3D spatial resolution, and reduced photodamage, meanwhile exogenous nanoprobes are usually needed for non-signal target cell analysis. Here, we report in-vivo, simultaneous 3D labeling and imaging of potato cell structures using plasmonic nanoprobe-assisted multimodal NLO microscopy. Experimental results show that the complete cell structure could be imaged by the combination of second-harmonic generation (SHG) and two-photon luminescence (TPL) when noble metal silver or gold ions are added. In contrast, without noble metal ion solution, no NLO signals from the cell wall could be acquired. The mechanism can be attributed to noble metal nanoprobes with strong nonlinear optical responses formed along the cell walls via a femtosecond laser scan. During the SHG-TPL imaging process, noble metal ions that cross the cell wall could be rapidly reduced to plasmonic nanoparticles by fs laser and selectively anchored onto both sides of the cell wall, thereby leading to simultaneous 3D labeling and imaging of potato cells. Compared with traditional labeling technique that needs in-vitro nanoprobe fabrication and cell labeling, our approach allows for one-step, in-vivo labeling of plant cells, thus providing a rapid, cost-effective way for cellular structure construction and imaging.Comment: 18 pages, 5 figure

A Liouville theorem for the fractional Ginzburg-Landau equation

In this paper, we are concerned with a Liouville-type result of the nonlinear integral equation \begin{equation*} u(x)=\int_{\mathbb{R}^{n}}\frac{u(1-|u|^{2})}{|x-y|^{n-\alpha}}dy, \end{equation*} where $u: \mathbb{R}^{n} \to \mathbb{R}^{k}$ with $k \geq 1$ and $1<\alpha<n/2$. We prove that $u \in L^2(\mathbb{R}^n) \Rightarrow u \equiv 0$ on $\mathbb{R}^n$, as long as $u$ is a bounded and differentiable solution.Comment: 7 page