In this paper, a class of semi-linear elliptic equations involving Hardy-Sobolev critical exponents has been investigated. This problem comes from the consideration of standing waves in the anisotropic Schr\"{o}dinger equation, also it is very important in the field of hydrodynamics, glaciology, quantum field theory and statistical mechanics. By a detailed estimation for the extremum function and using Mountain Pass Lemma with $\left( PS\right) _{c}$ conditions, the existence of positive solutions has been obtained. On the other hand, by establishing Pohozaev-type identity and using the properties of Bessel function, the nonexistence of positive solution also has been obtained. These results are extensions of E. Jannelli's research (\cite[Theorem 1.A-1.C]{EJ}).

With the rapidly increasing of people’s purchasing power, the fast moving consumer goods (FMCG) industry is supposed to grow dramatically. In order to gain more market access and profile, it is important for the FMCG manufacturers and retailers to find the preferences and provincial characteristics of consumers, to develop more suitable goods distribution strategy. Based on retails marketing data with geographic characteristics, this paper proposes a new combination of geography methods to solve the problems in distribution of FMCG. Via multiple K-means clustering and cross validation of KNN half off, the mesoscopic sales features are extracted through the classification of retails, which can indirectly grasp the consumer behavior characteristics. Based on space division and Moran’ I spatial autocorrelation arithmetic, two strategies are developed to satisfy consumer’s needs and promote sales, including conservative and positive strategies. According to our analysis, the total sales volume of the regions will increase by 5.1% and 10.3%. This study can be applied to the provide purchase strategies for FMCG retails according to their locations. The research can explore the consumption potential of different areas, thus improving the profile of retails and the development of economy in more mesoscopic scale.

No-till and cereal-legume intercropping have been recognized as favorable cropping practices to increase crop yields while maintaining soil quality in arid and semiarid environment, but the bio-logical mechanisms are poorly understood. The present study was to determine the response of soil properties, enzyme activities, and microbial community diversity and composition in mono- and inter-cropping under conventional and no-tillage conditions. We initiated a field experiment in Wuwei, a typical arid area of China, in 2014. Soil was sampled in August 2022 and, yields, soil properties, enzyme activities, and the microbial community diversity and composition were de-termined in the maize and pea strips in inter- and mono-cropping systems. Results revealed that the maize and pea strips in the no-till intercropping significantly increased yields, total and or-ganic carbon stocks, decreased NO3--N, and obtained the highest total and organic P in the soil. The α- and β-diversity of archaea and eukaryotes were significantly affected by planting patterns, while α- and β-diversity of the bacterial community were significantly affected by tillage practices. Both no-tillage and intercropped maize significantly increased the abundance of archaea phylum Thaumarchaeota and bacterial phylum Nitrospirae, benefiting nitrogen fixation of intercropped pea from the atmosphere under the no-tillage cereal/legume intercropping. No-till intercropping was conducive to the accumulation of organic carbon, while decreasing the abundance of Prote-obacteria, Acidobacteria, and Verrucomicrobia. Limited soil enzyme activities (ACP, ALP, DP, NAG, BG, AG, CB) led to decreases in organic carbon turnover and utilization. Intercropping al-tered soil microbial community diversity and composition due to changes in soil properties and enzyme activities. These findings suggest that no-tilled cereal-legume intercropping is a sustaina-ble cropping practice for improving soil properties and enhancing microbial (archaea, bacterial, Eukaryota) diversity, but the long-term persistence is not conducive to rapid turnover of soil nu-trients due to limited enzyme activities.