The embedding theorem in limiting cases and its applications in singular perturbation

In 1980, Brézis^[6], using the technique of dividing the total space into two parts, proved the embedding theorem of limiting case which is very important in applications. In 1982, Ding Xiaqi improved the proof given in [6], by using of the technique of dividing the total space into three parts. In this paper, using the technique of dividing the total space into three parts, the author proves uniformly the results obtained by Ding^[3,4], and gives an embedding theorem of limiting case including (Lemma 2.2). And he also gives two kinds of examples, applying the embedding theorems (limiting case and non-limiting case) and the interpolation theorems. These examples are the singular perturbation problems in the sense of Lions^[1] (for the definition of singular perturbation, see [1], Introduction). But the singular solutionU_e converges uniformly to the limit solution (degenerate)U, ase0.