Trudinger-Moser type inequalities for weighted Sobolev spaces involving fractional dimensions

Abstract

We derive sharp Trudinger-Moser inequalities for weighted Sobolev spaces and prove the existence of extremal functions. The inequalities we obtain here extend for fractional dimensions the classical results in the radial case. The main ingredient used in our arguments reveals a new proof of a result due to J. Moser for which we give an improved version.

On a weighted Trudinger-Moser inequality in RN

“…For the second part, we are going to do the changing of variable as in [15]. We define w(t) = ω 1 α+1…”

The sharp Poincaré–Sobolev type inequalities in the hyperbolic spaces Hn

“…In [24], de Oliveira and doÓ prove the following sharp Moser-Trudinger inequality involving the measure λ θ : suppose 0 < R < ∞ and α ≥ 2, θ ≥ 1, then (1). It is easy to see that D α,θ (R) = D α,θ R θ .…”

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.…”