Abstract
ABSTRACTLet be the unit ball in , be the standard Sobolev space. We consider an improved Trudinger–Moser inequality involving -norm. Denote the first eigenvalue of the n-Laplacian operator. We prove that for any p>1 and any , there holds where , is the area of the unit sphere in ; moreover, extremal functions for such inequalities exist. This is based on a method of blow-up analysis. The case was already solved by Yang [A sharp form of Moser-Trudinger inequality in high dimension. J Funct Anal. 2006;239:100–126] and Zhu [Improved Moser-Trudinger inequality involving norm in n dimensions. Adv Nonlinear Stud. 2014;14:273–293].
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