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This paper consider that the following semilinear elliptic equation

0.1 begin{document}$ begin{equation} left{ begin{array}{ll} -Delta u = u^{q(x)-1}, & {mbox{in}} B_1, u>0, & {mbox{in}} B_1, u = 0, & {mbox{in}} partial B_1, end{array} right. end{equation} $end{document}

where $ B_1 $ is the unit ball in $ mathbb{R}^N(Ngeq 3) $, $ q(x) = q(|x|) $ is a continuous radial function satifying $ 2leq q(x) < 2^* = frac{2N}{N-2} $ and $ q(0) > 2 $. Using variational methods and a priori estimate, the existence of a positive radial solution for (0.1) is obtained.

In this paper, for a regularized fuzzy system, a generalization of the direct Lyapunov method is adapted on the base of matrix-valued Lyapunov-like functions. First, the new concept of a regularization scheme for fuzzy systems is discussed and the matrix-valued Lyapunov function technique is introduced. Then, sufficient conditions are established for the boundedness and stability of the equilibrium set of solutions of the regularized fuzzy system of differential equations. Scalar and vector Lyapunov-type functions are used based on an auxiliary matrix-valued function. Finally, a discussion is offered for the future directions of the proposed approach. Since the strategies for the analysis of the stability of fuzzy models are very important in numerous aspects, we expect that our results will inspire researchers to develop the introduced concept.

The discrete element is an important tool for vibration compaction simulation from the microscopic viewpoint. The irregular particle model was established by the disc filling method, and the linear contact model with anti-rolling was selected to reflect the contact characteristics between the particles, so as to establish the simulation model of subgrade vibratory compaction. Based on this model, the stress characteristics of the area below the center of the vibrating wheel and the surface area of the soil were studied, and the principle of vibratory compaction was discussed. The results show that the distribution of vertical stresses below the center of drum basically presents a decreasing trend in the depth range during vibration, with the stress amplitude of the lower structure increasing and the stress magnitude of the upper structure decreasing. The distribution of horizontal stresses in the area below the center of the vibrating wheel is similar to the stress distribution in the splitting test. The soil at the surface has an obvious pushing and squeezing effect, and the transmission distance of horizontal stresses is larger than that of vertical stresses. The soil at the surface is pushed and the horizontal stresses are transmitted at a greater distance than the vertical stresses, which, together with a certain degree of shear effect, causes a certain uplift deformation of the soil around the vibrating wheel. In general, the vibration compaction process is relatively consistent with the theory of repeated loading and the theory of alternating shear strain.

With the continuous development of science and technology (especially computational devices with powerful computing capabilities), the image generation technology based on deep learning has also made significant achievements. Most cross-modal technologies based on deep learning can generate information from text into images, which has become a hot topic of current research. Text-to-image (T2I) synthesis technology has applications in multiple fields of computer vision, such as image enhancement, artificial intelligence painting, games and virtual reality. The T2I generation technology using generative adversarial networks can generate more realistic and diverse images, but there are also some shortcomings and challenges, such as difficulty in generating complex backgrounds. This review will be introduced in the following order. First, we introduce the basic principles and architecture of basic and classic generative adversarial networks (GANs). Second, this review categorizes T2I synthesis methods into four main categories. There are methods based on semantic enhancement, methods based on progressive structure, methods based on attention and methods based on introducing additional signals. We have chosen some of the classic and latest T2I methods for introduction and explain their main advantages and shortcomings. Third, we explain the basic dataset and evaluation indicators in the T2I field. Finally, prospects for future research directions are discussed. This review provides a systematic introduction to the basic GAN method and the T2I method based on it, which can serve as a reference for researchers.