Equivalence of critical and subcritical sharp Trudinger–Moser–Adams inequalities

Abstract

Sharp Trudinger-Moser inequalities on the first order Sobolev spaces and their analogous Adams inequalities on high order Sobolev spaces play an important role in geometric analysis, partial differential equations and other branches of modern mathematics. Such geometric inequalities have been studied extensively by many authors in recent years and there is a vast literature. There are two types of such optimal inequalities: critical and subcritical sharp inequalities, both are with best constants. Critical sharp inequalities are under the restriction of the full Sobolev norms for the functions under consideration, while the subcritical inequalities are under the restriction of the partial Sobolev norms for the functions under consideration. There are subtle differences between these two type of inequalities. Surprisingly, we prove in this paper that these critical and subcritical Trudinger-Moser and Adams inequalities are actually equivalent. Moreover, we also establish the asymptotic behavior of the supremum for the subcritical Trudinger-Moser and Adams inequalities on the entire Euclidean spaces (Theorem 1.1 and Theorem 1.3) and provide a precise relationship between the supremums for the critical and subcritical Trudinger-Moser and Adams inequalities (Theorem 1.2 and Theorem 1.4). Since the critical Trudinger-Moser and Adams inequalities can be easier to prove than subcritical ones in some occasions, and more difficult to establish in other occasions, our results and the method suggest a new approach to both the critical and subcritical Trudinger-Moser and Adams type inequalities.

Existence and nonexistence of extremals for critical Adams inequalities in R4 and Trudinger-Moser inequalities in R2Chen1, Lu2, Zhu3 2020Advances in MathematicsSelf Cite22040View full textAdd to dashboardCite

“…We finally remark that there is some recent development on the existence and nonexistence of extremal functions for subcritical Trudinger-Moser inequalities established by Lam, Lu and Zhang [21] using the equivalence and identities between the supremums for the critical and subcritical Trudinger-Moser inequalities in R n established by the same authors in [22]. For subcritical Adams inequalities on the entire space, the existence of extremal functions has been proved by Chen, Lu and Zhang [7].…”

Optimizers for the singular Trudinger–Moser inequalities in the critical case in

“…The equivalence of the sharp subcritical and critical Trudinger–Moser inequalities and the asymptotic behavior of were pointed out in for , and in for the general case. Moreover, in , the authors proved the following more general results: Theorem A if and only if () or ( ). The constant is sharp.…”

“…These two results are often called as the sharp subcritical and critical Trudinger-Moser inequalities, correspondingly. The equivalence of the sharp subcritical and critical Trudinger-Moser inequalities and the asymptotic behavior of (⋅) were pointed out in [6] for = 2, and in [15] for the general case. Moreover, in [15], the authors proved the following more general results:…”

Vanishing-concentration-compactness alternative for the Trudinger–Moser inequality in ℝN

“…In a very recent paper, Lam, Lu and Zhang [15] proved that, when d N,αN (a, b) < +∞, the exponent α N is sharp for the corresponding Trudinger-Moser inequality and it is not affected by the values of a and b. in the sense that…”