A new perturbation to a critical elliptic problem with a variable exponent Dedicated to Professor Daomin Cao on the Occasion of His Sixtieth Birthday

In this paper,we study the following critical elliptic problem with a variable exponent:{-Δu=u^(p+ϵa(x))inΩ,u>0 inΩ,u=0 on∂Ω,where a(x)2∈C^(2)(Ω),p=N+2/N-2,∈>0,andΩis a smooth bounded domain in R^(N)(N>4).We show that for∈small enough,there exists a family of bubble solutions concentrating at the negative stable critical point of the function a(x).This is a new perturbation to the critical elliptic equation in contrast to the usual subcritical or supercritical perturbation,and gives the first existence result for the critical elliptic problem with a variable exponent.

Exploratory Analysis on Dictated Handwriting Samples

Dictated handwriting samples are widely used in practice due to their simplicity,convenience,and practicality.However,dictation is typically listed as one of the many collection methods in textbooks and monographs,and there is usually no separate section focusing on dictated handwriting samples.Therefore,further study of dictated handwriting samples will have important practical significance.Consideration of the definition,existing problems,collection techniques,and critical aspects of dictated handwriting samples willsupport investigators and document examiners in their professional abilities and contribute to the theoretical system of document examination.In this article,an exploratory analysis will be conducted and ideas about dictated handwriting samples will be shared,including the definition of dictated samples,their relationship to experimental samples,practical problems,feasible collection methods,and some critical points that require special attention.Dictation is widely used but problematic because of a lack of quantity and low levels of comparability.Those difficulties are mainly caused by a lack of theoreticalstudy and understanding of the requirements and collection techniques of dictated handwriting samples among first‑line investigators.Dictated samples should be collected based on subjective and objective conditions of formation with aims to improve comparability and five similarities.Further studies are needed to improve the theoretical system and practical use of dictated samples so that they can contribute to successfully reaching conclusions in investigations.