Weighted Trudinger-Moser inequalities and associated Liouville type equations

Abstract

We discuss some Trudinger-Moser inequalities with weighted Sobolev norms. Suitable logarithmic weights in these norms allow an improvement in the maximal growth for integrability, when one restricts to radial functions. The main results concern the application of these inequalities to the existence of solutions for certain mean-field equations of Liouville-type. Sharp critical thresholds are found such that for parameters below these thresholds the corresponding functionals are coercive and hence solutions are obtained as global minima of these functionals. In the critical cases the functionals are no longer coercive and solutions may not exist. We also discuss a limiting case, in which the allowed growth is of double exponential type. Surprisingly, we are able to show that in this case a local minimum persists to exist for critical and also for slightly supercritical parameters. This allows to obtain the existence of a second (mountain-pass) solution, for almost all slightly supercritical parameters, using the Struwe monotonicity trick. This result is in contrast to the non-weighted case, where positive solutions do not exist (in star-shaped domains) in the critical and supercritical case.

N- Laplacian problems with critical double exponential nonlinearities

“…where the weight ω(x) is defined in (4), the function f (x, u) is continuous in B × R and has critical double exponential growth, which behaves like exp{e α|u|…”

“…N −1 and ω N −1 is the surface area of the unit ball in R N . Recently, the influence of weights on limiting inequalities of Trudinger-Moser type has been studied, for example, see [3,9,4,5,14,15]. If ω ∈ L 1 (Ω) is a non-negative function, we introduce the weighted Sobolev space…”

A class of Schrödinger elliptic equations involving supercritical exponential growth

“…In order to extend equations (1.1), we will study Schrödinger equations involving a diffusion operator (see [10,12,32,38,39] among others). Let B 1 be the unit ball centered at the origin in R 2 and H 1 0,rad (B 1 , w) be the subspace of the radially symmetric functions in the closure of C ∞ 0 (B 1 ) with respect to the norm…”

Singular weighted sharp Trudinger-Moser inequalities defined on $ \mathbb{R}^N $ and applications to elliptic nonlinear equations

“…the existence of extremal functions), we can refer to [30,32,36]. Now, concerning Trudinger-Moser inequalities defined on weighted Sobolev spaces, we can for example cite [1,5,6,7,14,15,16,17,23,25,26,28,36]. The majority of those works considered (directly or through a rearrangement) the restriction to radial functions.…”