Various functional inequalities and their applications to the variational problems

我们的主要研究目的在于研究Sobolev型不等式及其相应的变分问题。我们特别关注了Trudinger-Moser型不等式、Gagliardo-Nirenberg型插值不等式等中出现的临界嵌入。在本基金的一段时间里，我们研究了与Adachi-Tanaka（1999）得到的尺度不变形式的Trudinger-Moser型不等式相关的最大化器的存在性，并实际证明了最大化器的存在性。另一方面，已知整个空间上的相似型Trudinger-Moser不等式不满足标度不变性。Ishiwata（2010）证明了该不等式的最大化者的存在在很大程度上取决于维度。基于这一事实，我们考虑了变分结构，并阐明了尺度不变性与存在最大化器之间的关系。

Our main purpose of the research lies in studying Sobolev type inequalities and the corresponding variational problems. Especially, we concern the critical embeddings which appear in Trudinger-Moser type inequalities, Gagliardo-Nirenberg type interpolation inequalities and so on. In a period of this fund, we investigated the existence of maximizer associated with the Trudinger-Moser type inequalities of the scaling invariant form obtained by Adachi-Tanaka in 1999, and actually proved the existence of a maximizer. On the other hand, it was known that the similar type Trudinger-Moser inequalities on the whole space which do not satisfy the scaling invariance. Ishiwata (2010) proved that the existence of maximizers for this inequality heavily depends on the dimension. Based on this fact, we considered the variational structure and clarified the relation between the scaling invariance and the existence of a maximizer.