EQUIVALENT MOSER TYPE INEQUALITIES IN R 2 AND THE ZERO MASS CASE (original) (raw)

4 Equivalent Moser Type Inequalities in R2 and the Zero Mass Case

2016

We first investigate concentration and vanishing phenomena concerning Moser type inequalities in the whole plane which involve complete and reduced Sobolev norms. In particular we show that the critical Ruf inequality is equivalent to an improved version of the subcritical Adachi-Tanaka inequality which we prove to be attained. Then, we consider the limiting space D 1,2 (R 2), completion of smooth compactly supported functions with respect to the Dirichlet norm ∇ ⋅ 2 , and we prove an optimal Lorentz-Zygmund type inequality with explicit extremals and from which can be derived classical inequalities in H 1 (R 2) such as the Adachi-Tanaka inequality and a version of Ruf's inequality.

On Trudinger–Moser type inequalities involving Sobolev–Lorentz spaces

Annali di Matematica Pura ed Applicata, 2009

Generalizations of the Trudinger-Moser inequality to Sobolev-Lorentz spaces with weights are considered. The weights in these spaces allow for the addition of certain lower order terms in the exponential integral. We prove an explicit relation between the weights and the lower order terms; furthermore, we show that the resulting inequalities are sharp, and that there are related phenomena of concentration-compactness.

On Trudinger-Moser type inequalities involving

2008

OPTIMAL SOBOLEV TYPE INEQUALITIES IN LORENTZ SPACES

It is well known that classical Sobolev's embeddings may be improved within the framework of Lorentz spaces. However, the value of the best possible constants in the corresponding inequalities is known in just one case by the work of A. Alvino. Here, we determine optimal constants for the embedding of the space D 1,p (R n), 1 < p < n, into the whole Lorentz space scale L p * ,q (R n), p ≤ q ≤ ∞; including the case of Alvino q = p of which we give a new proof. We finally exhibit extremal functions for the embedding inequalities by solving related elliptic problems.

Moser-Type Inequalities in Lorentz Spaces*

Potential Analysis, 1996

Moser-type estimates for functions whose gradient is in the Lorentz space L(n, q), 1 ~< q ~< ~, are given. Similar results are obtained for solutions u E H~ ofAu = (fOx,, where A is a linear elliptic second order differential operator and [fl ~ L(n, q), 2 <~ q <~ oo.

A note on Sobolev inequalities and limits of Lorentz spaces

Contemporary Mathematics, 2007

Motivated by the theory of Sobolev embeddings we shall present a new way to obtain L ∞ estimates by means of taking limits of Lorentz spaces (*extrapolation*). Although our result is independent from the theory of embeddings we thought it would be worthwhile to present rather succinctly the issues that motivated us. We refer to other papers in these proceedings for more complete and detailed accounts of the relevant theory of embeddings.

Remarks on the Moser–Trudinger inequality

Advances in Nonlinear Analysis, 2013

In this article, we study the existence of multiple solutions to a generalized () ⋅ p-Laplace equation with two parameters involving critical growth. More precisely, we give sufficient "local" conditions, which mean that growths between the main operator and nonlinear term are locally assumed for () ⋅ p-sublinear, () ⋅ p-superlinear, and sandwich-type cases. Compared to constant exponent problems (e.g., p-Laplacian and () p q ,-Laplacian), this characterizes the study of variable exponent problems. We show this by applying variants of the mountain pass theorem for () ⋅ p-sublinear and () ⋅ p-superlinear cases and constructing critical values defined by a minimax argument in the genus theory for sandwich-type case. Moreover, we also obtain a nontrivial nonnegative solution for sandwich-type case changing the role of parameters. Our work is a generalization of several existing works in the literature.

EQUIVALENT MOSER TYPE INEQUALITIES IN R 2 AND THE ZERO MASS CASE (original) (raw)

Related papers