In this paper we present a detailed study of critical embeddings of weighted Sobolev spaces into weighted Orlicz spaces of exponential type for weights of monomial type. More precisely, we give an alternative proof of a recent result by N. Lam [NoDEA 24(4), 2017] showing the optimality of the constant in the Trudinger–Moser inequality. We prove a Poincaré inequality for this class of weights. We show that the critical embedding is optimal within the class of Orlicz target spaces. Moreover, we prove that it is not compact, and derive a corresponding version of P.-L. Lions’ principle of concentrated compactness.

中文翻译：

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关于单项权重的Trudinger-Moser不等式的更多见解

在本文中，我们对加权Sobolev空间的临界嵌入到单项式权重的指数型加权Orlicz空间进行了详细研究。更准确地说，我们给出了N. Lam [NoDEA 24（4），2017]最近的结果的替代证明，该结果显示了Trudinger-Moser不等式中常数的最优性。我们证明了此类权重的庞加莱不等式。我们表明，临界嵌入在Orlicz目标空间的类别内是最佳的。此外，我们证明它不是紧凑的，并推导了P.-L的相应版本。狮子会的紧凑性原则。