On Trudinger–Moser type inequalities with logarithmic weights

Weighted Trudinger - Moser Inequalities and Applications

“…In fact, by a radial lemma, see [17]), for such ψ it holds that |ψ(t)| ≤ t 1/γ . With this estimate the subcritical case can be trivially proved.…”

“…The proof is quite technical and we refer the reader to [17] for details. The idea is to transform the problem (which is one-dimensional by the radial symmetry) to an integral inequality on the half-line.…”

“…But again, the embedding does not go into L ∞ , but we nd a critical growth of double exponential type, as described in the following Theorem 5. [17] …”

“…Indeed, translating Moser's functions to a region in which the presence of the weight is not substantial (see [17]) one concludes that the presence of the weight has no eect for increasing the range of the inequality.…”

Extremal function for Moser–Trudinger type inequality with logarithmic weight

“…The subspace of radial functions in H 1 0 (w 0 , B) is denoted by H 1 0,rad (w 0 , B). The following theorem by Calanchi-Ruf in [5] generalizes the Moser-Trudinger inequality for balls. The work of this paper is based on this generalization.…”