# Sharp singular Adams inequalities in high order Sobolev spaces

@article{Lam2011SharpSA, title={Sharp singular Adams inequalities in high order Sobolev spaces}, author={Nguyen Huong Lam and Guozhen Lu}, journal={arXiv: Analysis of PDEs}, year={2011} }

In this paper, we prove a version of weighted inequalities of exponential type for fractional integrals with sharp constants in any domain of finite measure in $\mathbb{R}^{n}$. Using this we prove a sharp singular Adams inequality in high order Sobolev spaces in bounded domain at critical case. Then we prove sharp singular Adams inequalities for high order derivatives on unbounded domains. Our results extend the singular Moser-Trudinger inequalities of first order in \cite{Ad2, R, LR, AdY} to…

## 35 Citations

Sharpened Adams Inequality and Ground State Solutions to the Bi-Laplacian Equation in ℝ4

- MathematicsAdvanced Nonlinear Studies
- 2018

Abstract In this paper, we establish a sharp concentration-compactness principle associated with the singular Adams inequality on the second-order Sobolev spaces in ℝ 4 {\mathbb{R}^{4}} . We also…

A new approach to sharp Moser–Trudinger and Adams type inequalities: A rearrangement-free argument

- Mathematics
- 2013

Sharp Affine and Improved Moser–Trudinger–Adams Type Inequalities on Unbounded Domains in the Spirit of Lions

- Mathematics
- 2017

The purpose of this paper is threefold. First, we prove sharp singular affine Moser–Trudinger inequalities on both bounded and unbounded domains in $${\mathbb {R}}^{n}$$Rn. In particular, we will…

The sharp Adams type inequalities in the hyperbolic spaces under the Lorentz-Sobolev norms.

- Mathematics
- 2020

Let $2\leq m < n$ and $q \in (1,\infty)$, we denote by $W^mL^{\frac nm,q}(\mathbb H^n)$ the Lorentz-Sobolev space of order $m$ in the hyperbolic space $\mathbb H^n$. In this paper, we establish the…

Sharp weighted Trudinger–Moser–Adams inequalities on the whole space and the existence of their extremals

- MathematicsCalculus of Variations and Partial Differential Equations
- 2019

Though there have been extensive works on the existence of maximizers for sharp first order Trudinger–Moser inequalities, much less is known for that of the maximizers for higher order Adams’…

Sharp Singular Trudinger-Moser-Adams Type Inequalities with Exact Growth

- Mathematics
- 2015

The main purpose of this paper is two fold. On the one hand, we review some recent progress on best constants for various sharp Moser-Trudinger and Adams inequalities in Euclidean spaces…

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

- Mathematics
- 2015

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…

SHARP ADAMS TYPE INEQUALITIES 3

- Mathematics
- 2011

The main purpose of our paper is to prove sharp Adams-type inequalities in unbounded domains of R for the Sobolev space W n m (R) for any positive integer m less than n. Our results complement those…

Fractional Adams–Moser–Trudinger type inequality on Heisenberg group

- MathematicsNonlinear Analysis
- 2020

## References

SHOWING 1-10 OF 75 REFERENCES

An improved Hardy-Sobolev inequality and its application

- Mathematics
- 2001

For\Omega \subset $IR^n$,n\geq 2, a bounded domain, and for 1 < p < n, we improve the Hardy-Sobolev inequality, by adding a term with a singular weight of the type \frac{1}{log(1/|x|)}$^2$ . We show…

EXISTENCE OF NONTRIVIAL SOLUTIONS TO POLYHARMONIC EQUATIONS WITH SUBCRITICAL AND CRITICAL EXPONENTIAL GROWTH

- Mathematics
- 2012

The main purpose of this paper is to establish
the existence of nontrivial solutions to semilinear polyharmonic
equations with exponential growth at the subcritical or critical level. This growth…

SHARP ADAMS-TYPE INEQUALITIES IN R

- Mathematics
- 2012

Adams’ inequality for bounded domains Ω ⊂ R4 states that the supremum of ∫ Ω e 32πu dx over all functions u ∈ W 2, 2 0 (Ω) with ‖Δu‖2 ≤ 1 is bounded by a constant depending on Ω only. This bound…

On a version of Trudinger–Moser inequality with Möbius shift invariance

- Mathematics
- 2009

The paper raises a question about the optimal critical nonlinearity for the Sobolev space in two dimensions, connected to loss of compactness, and discusses the pertinent concentration compactness…

Blow-up Analysis in Dimension 2 and a Sharp Form of Trudinger–Moser Inequality

- Mathematics
- 2005

Abstract This paper deals with an improvement of the Trudinger–Moser inequality associated to the embedding of the standard Sobolev space into Orlicz spaces for Ω a smooth bounded domain in ℝ2. The…