On the spectrum of a class of differential operators and embedding theorems

The author considers the embedding problem of weighted Sobolev spacesH _p ⁿ in weightedL _s spacesL _s,r , and some sufficient conditions and necessary conditions are given, when weight functions satisfy certain conditions. The author uses the results obtained to the qualitative analysis of the spectrum of 2n-order weighted differential operator, and gives some sufficient conditions and necessary conditions to ensure that the spectrum is discrete. Supported by the National Natural Science Fundation of China and the Natural Science Foundation of Inner Mongolia.