More insights into the Trudinger-Moser inequality with monomial weight. (English) Zbl 1469.46031
Summary: 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, Nonlinear Differ. Equ. Appl. 24, No. 4, Paper No. 39, 21 p. (2017; Zbl 1375.35012)]
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.
MSC:
| 46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |
| 46E30 | Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) |
| 35A23 | Inequalities applied to PDEs involving derivatives, differential and integral operators, or integrals |
| 26D15 | Inequalities for sums, series and integrals |
Citations:
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