Ground state solutions to a coupled nonlinear logarithmic Hartree system

In this paper, we study the following coupled nonlinear logarithmic Hartree system \begin{align*} \left\{ \displaystyle \begin{array}{ll} \displaystyle -\Delta u+ \lambda_1 u =\mu_1\left( -\frac{1}{2\pi}\ln(|x|) \ast u^2 \right)u+\beta \left( -\frac{1}{2\pi}\ln(|x|) \ast v^2 \right)u, & x \in ~ \mathbb R^2, \vspace{.4cm}\\ -\Delta v+ \lambda_2 v =\mu_2\left( -\frac{1}{2\pi}\ln(|x|) \ast v^2 \right)v +\beta\left( -\frac{1}{2\pi}\ln(|x|) \ast u^2 \right)v, & x \in ~ \mathbb R^2, \end{array} \right.\hspace{1cm} \end{align*} where $\beta, \mu_i, \lambda_i \ (i=1,2)$ are positive constants, $\ast$ denotes the convolution in $\mathbb R^2$. By considering the constraint minimum problem on the Nehari manifold, we prove the existence of ground state solutions for $\beta>0$ large enough. Moreover, we also show that every positive solution is radially symmetric and decays exponentially

Existence of nontrivial solutions for critical biharmonic equations with logarithmic term

In this paper, we consider the existence of nontrivial solutions to the following critical biharmonic problem with a logarithmic term \begin{equation*} \begin{cases} \Delta^2 u=\mu \Delta u+\lambda u+|u|^{2^{**}-2}u+\tau u\log u^2, \ \ x\in\Omega, u|_{\partial \Omega }=\frac{\partial u}{\partial n}|_{\partial\Omega}=0, \end{cases} \end{equation*} where $\mu,\lambda,\tau \in \mathbb{R}$, $|\mu|+|\tau|\ne 0$, $\Delta ^2=\Delta \Delta$ denotes the iterated N-dimensional Laplacian, $\Omega \subset \mathbb{R}^{N}$ is a bounded domain with smooth boundary $\partial \Omega$, $2^{**}=\frac{2N}{N-4}(N\ge5)$ is the critical Sobolev exponent for the embedding $H_{0}^{2}(\Omega)\hookrightarrow L^{2^{**}}(\Omega)$ and $H_0^2 (\Omega )$ is the closure of $C_0^ \infty (\Omega )$ under the norm $|| u ||:=(\int_{\Omega}|\Delta u|^2)^\frac{1}{2}$. The uncertainty of the sign of $s\log s^2$ in $(0,+\infty)$ has some interest in itself. To know which of the three terms $\mu \Delta u$, $\lambda u$ and $\tau u \log u^2$ has a greater influence on the existence of nontrivial weak solutions, we prove the existence of nontrivial weak solutions to the above problem for $N\ge5$ under some assumptions of $\lambda, \mu$ and $\tau$

The existence of positive solution for an elliptic problem with critical growth and logarithmic perturbation

We consider the existence and nonexistence of positive solution for the following Br\'ezis-Nirenberg problem with logarithmic perturbation: \begin{equation*} \begin{cases} -\Delta u={\left|u\right|}^{{2}^{\ast }-2}u+\lambda u+\mu u\log {u}^{2} &x\in \Omega, \quad \;\:\, u=0& x\in \partial \Omega, \end{cases} \end{equation*} where $\Omega$ $\subset$ $\R^N$ is a bounded smooth domain, $\lambda, \mu \in \R$, $N\ge3$ and ${2}^{\ast }:=\frac{2N}{N-2}$ is the critical Sobolev exponent for the embedding $H^1_{0}(\Omega)\hookrightarrow L^{2^\ast}(\Omega)$. The uncertainty of the sign of $s\log s^2$ in $(0, +\infty)$ has some interest in itself. We will show the existence of positive ground state solution which is of mountain pass type provided $\lambda\in \R, \mu>0$ and $N\geq 4$. While the case of $\mu<0$ is thornier. However, for $N=3,4$ $\lambda\in (-\infty, \lambda_1(\Omega))$, we can also establish the existence of positive solution under some further suitable assumptions. And a nonexistence result is also obtained for $\mu<0$ and $-\frac{(N-2)\mu}{2}+\frac{(N-2)\mu}{2}\log(-\frac{(N-2)\mu}{2})+\lambda-\lambda_1(\Omega)\geq 0$ if $N\geq 3$. Comparing with the results in Br\'ezis, H. and Nirenberg, L. (Comm. Pure Appl. Math. 1983), some new interesting phenomenon occurs when the parameter $\mu$ on logarithmic perturbation is not zero

Neural Residual Radiance Fields for Streamably Free-Viewpoint Videos

The success of the Neural Radiance Fields (NeRFs) for modeling and free-view rendering static objects has inspired numerous attempts on dynamic scenes. Current techniques that utilize neural rendering for facilitating free-view videos (FVVs) are restricted to either offline rendering or are capable of processing only brief sequences with minimal motion. In this paper, we present a novel technique, Residual Radiance Field or ReRF, as a highly compact neural representation to achieve real-time FVV rendering on long-duration dynamic scenes. ReRF explicitly models the residual information between adjacent timestamps in the spatial-temporal feature space, with a global coordinate-based tiny MLP as the feature decoder. Specifically, ReRF employs a compact motion grid along with a residual feature grid to exploit inter-frame feature similarities. We show such a strategy can handle large motions without sacrificing quality. We further present a sequential training scheme to maintain the smoothness and the sparsity of the motion/residual grids. Based on ReRF, we design a special FVV codec that achieves three orders of magnitudes compression rate and provides a companion ReRF player to support online streaming of long-duration FVVs of dynamic scenes. Extensive experiments demonstrate the effectiveness of ReRF for compactly representing dynamic radiance fields, enabling an unprecedented free-viewpoint viewing experience in speed and quality.Comment: Accepted by CVPR 2023. Project page, see https://aoliao12138.github.io/ReRF

Multifunctional actuation systems responding to chemical gradients

The ability to manipulate the movement of surface microstructures is essential for the development of dynamic, responsive materials. We demonstrate that in addition to bulk actuation upon drying, a unique type of highly localized, directional actuation can be achieved when microstructures embedded in pH responsive gel are exposed to pH gradients. Theory and modelling elucidates the underlying mechanism behind this novel approach to inducing responsive actuation.Chemistry and Chemical BiologyEngineering and Applied Science

Nontrivial solutions to the p-harmonic equation with nonlinearity asymptotic to |t|p–2t at infinity

We consider the following p-harmonic proble

Role of video games in college life in China

Video games have become an increasingly integral part of Chinese society. Together with Beijing University of Chemical Technology, we utilized surveys and interviews from both students and professors to understand the impacts of video games on a college campus in China. Despite the perceived generational gap both groups showed positive views on video games. This work suggests that the use of video games in the classroom and events can be used to improve the social and academic lives of students