| Abstract: |
| We discuss the existence or non-existence of maximizers for Trudinger-Moser type inequalities of inhomogeneous type. Trudinger-Moser inequality was originally obtained by Trudinger and Moser in bounded domains as the limiting case of Sobolev embeddings. After that, several authors succeeded in extending the inequality to unbounded domains. Especially, Ruf (2005) and Li-Ruf (2008) established inhomogeneous Trudinger-Moser type inequalities in the whole space with best constants. In this talk, we will consider the existence and non-existence of these type inequalities. The difficulties come from a lack of compactness of the corresponding functional due to concentrating and vanishing phenomena. We shall give sufficient conditions on the exponents appearing in the inequality so that its maximizers exist or not. This is a joint work with Professor Norihisa Ikoma in Keio University and Professor Michinori Ishiwata in Osaka University. |
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