Finite Morse index solutions and asymptotics of weighted nonlinear elliptic equations

By introducing a suitable setting, we study the behavior of finite Morse index solutions of the equation -\{div} (|x|^\theta \nabla v)=|x|^l |v|^{p-1}v \;\;\; \{in $\Omega \subset \R^N \; (N \geq 2)$}, \leqno(1) where $p>1$, $\theta, l\in\R^1$ with $N+\theta>2$, $l-\theta>-2$, and $\Omega$ is a bounded or unbounded domain. Through a suitable transformation of the form $v(x)=|x|^\sigma u(x)$, equation (1) can be rewritten as a nonlinear Schr\"odinger equation with Hardy potential -\Delta u=|x|^\alpha |u|^{p-1}u+\frac{\ell}{|x|^2} u \;\; \{in $\Omega \subset \R^N \;\; (N \geq 2)$}, \leqno{(2)} where $p>1$, $\alpha \in (-\infty, \infty)$ and $\ell \in (-\infty,(N-2)^2/4)$. We show that under our chosen setting for the finite Morse index theory of (1), the stability of a solution to (1) is unchanged under various natural transformations. This enables us to reveal two critical values of the exponent $p$ in (1) that divide the behavior of finite Morse index solutions of (1), which in turn yields two critical powers for (2) through the transformation. The latter appear difficult to obtain by working directly with (2)

Coexistence states for systems of mutualist species

AbstractCoexistence states for a class of systems of mutualist species are obtained via bifurcation theory and monotone techniques

TET-GAN: Text Effects Transfer via Stylization and Destylization

Text effects transfer technology automatically makes the text dramatically more impressive. However, previous style transfer methods either study the model for general style, which cannot handle the highly-structured text effects along the glyph, or require manual design of subtle matching criteria for text effects. In this paper, we focus on the use of the powerful representation abilities of deep neural features for text effects transfer. For this purpose, we propose a novel Texture Effects Transfer GAN (TET-GAN), which consists of a stylization subnetwork and a destylization subnetwork. The key idea is to train our network to accomplish both the objective of style transfer and style removal, so that it can learn to disentangle and recombine the content and style features of text effects images. To support the training of our network, we propose a new text effects dataset with as much as 64 professionally designed styles on 837 characters. We show that the disentangled feature representations enable us to transfer or remove all these styles on arbitrary glyphs using one network. Furthermore, the flexible network design empowers TET-GAN to efficiently extend to a new text style via one-shot learning where only one example is required. We demonstrate the superiority of the proposed method in generating high-quality stylized text over the state-of-the-art methods.Comment: Accepted by AAAI 2019. Code and dataset will be available at http://www.icst.pku.edu.cn/struct/Projects/TETGAN.htm