An improvement for the Trudinger–Moser inequality and applications

In line with the Concentration–Compactness Principle due to P.-L. Lions [19], we study the lack of compactness of Sobolev embedding of W^1,n(Rⁿ)$W^{1,n}({\mathbb{R}}^n)$, n?2$n?2$, into the Orlicz space _LΦα$L_{{\Phi }_{\alpha }}$ determined by the Young function _Φα(s)${\Phi }_{\alpha }(s)$ behaving like e^{α|s|n/(n−1)}−1$e^{\alpha {|s|}^{n/(n-1)}}-1$ as |s|→+∞$|s|\to +\infty$. In the light of this result we also study existence of ground state solutions for a class of quasilinear elliptic problems involving critical growth of the Trudinger–Moser type in the whole space Rⁿ${\mathbb{R}}^n$.     

A Trudinger–Moser inequality for a conical metric in the unit ball

Archiv der Mathematik - In this note, we prove a Trudinger–Moser inequality for a conical metric in the unit ball. Precisely, let $${\mathbb {B}}$$ be the unit ball in $${\mathbb {R}}^N$$...     

Moser–Trudinger inequalities for singular Liouville systems

Mathematische Zeitschrift - In this paper we prove a Moser–Trudinger inequality for the Euler–Lagrange functional of general singular Liouville systems on a compact surface. We...     

Affine Moser–Trudinger and Morrey–Sobolev inequalities

An affine Moser–Trudinger inequality, which is stronger than the Euclidean Moser–Trudinger inequality, is established. In this new affine analytic inequality an affine energy of the gradient replaces the standard L ⁿ energy of gradient. The geometric inequality at the core of the affine Moser–Trudinger inequality is a recently established affine isoperimetric inequality for convex bodies. Critical use is made of the solution to a normalized version of the L ⁿ Minkowski Problem. An affine Morrey–Sobolev inequality is also established, where the standard L ^p energy, with p > n, is replaced by the affine energy.     

Concentration-compactness principles for Moser–Trudinger inequalities: new results and proofs

We are concerned with the best exponent in Concentration-Compactness principles for the borderline case of the Sobolev inequality. We present a new approach, which both yields a rigorous proof of the relevant principle in the standard case when functions vanishing on the boundary are considered, and enables us to deal with functions with unrestricted boundary values.     

A Trudinger–Moser inequality of Adimurthi–Druet type involving higher order eigenvalues

Archiv der Mathematik - Let $$Omega $$ be a smooth bounded domain in $${mathbb {R}}^2$$ and $$W_0^{1, 2}(Omega )$$ be the usual Sobolev space. Assume that $$0<lambda _1(Omega...     

Wiener–Ingham type inequality for Vilenkin groups and its application to harmonic analysis

An important trigonometric inequality essentially due to Wiener but later on made precise by Ingham concerning the lacunary trigonometric sums \(f(x)=\sum A_ke^{in_kx}\), where \(A_k\)’s are complex numbers, \(n_{-k}=-n_k\) and \(\{n_k\}\) satisfies the small gap condition \((n_{k+1}-n_k)\ge q\ge 1\) for \(k=0,1,2,\ldots \), says that if I is any subinterval of \([-\pi ,\pi ]\) of length \(|I|=2\pi (1+\delta )/q>2\pi /q\) then \(\sum |A_k|^2\le A_{\delta }|I|^{-1}\int _I|f|^2\), \(|A_k|\le A_{\delta }|I|^{-1}\int _I|f|\), wherein \(A_{\delta }\) depends only on \(\delta \). Such an inequality is proved here in the setting of the Vilenkin groups G. The inequality is then applied to generalize the Bernstěin, Szász and Ste?hkin type results concerning the absolute convergence of Fourier series on G.     

On the sharp constant for the weighted Trudinger–Moser type inequality of the scaling invariant form

In this article, we establish the weighted Trudinger–Moser inequality of the scaling invariant form including its best constant and prove the existence of a maximizer for the associated variational problem. The non-singular case was treated by Adachi and Tanaka (1999) [1] and the existence of a maximizer is a new result even for the non-singular case. We also discuss the relation between the best constants of the weighted Trudinger–Moser inequality and the Caffarelli–Kohn–Nirenberg inequality in the asymptotic sense.     

Improved Singular Moser–Trudinger Inequalities and Their Extremal Functions

Potential Analysis - In this paper, we prove several improvements for the sharp singular Moser–Trudinger inequality. We first establish an improved singular Moser–Trudinger inequality...     

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

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 framework. We study properties of the improved version of the Trudinger–Moser inequality on the open unit disk ${B\subset\mathbb R^2}$ , recently proved by Mancini and Sandeep [g], (Arxiv 0910.0971). Unlike the original Trudinger–Moser inequality, this inequality is invariant with respect to the Möbius automorphisms of the unit disk, and as such is a closer analogy of the critical nonlinearity ${\int |u|^{2^*}}$ in the higher dimension than the original Trudinger–Moser nonlinearity.