A Trudinger-Moser inequality in a weighted Sobolev space and applications (original) (raw)

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

Proceedings of the American Mathematical Society, 2014

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.

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.

Sharp Moser–Trudinger inequalities for the Laplacian without boundary conditions

Journal of Functional Analysis, 2012

We derive a sharp Moser-Trudinger inequality for the borderline Sobolev imbedding of W 2,n/2 (B n) into the exponential class, where B n is the unit ball of R n. The corresponding sharp results for the spaces W d,n/d 0 (Ω) are well known, for general domains Ω, and are due to Moser and Adams. When the zero boundary condition is removed the only known results are for d = 1 and are due to Chang-Yang, Cianchi and Leckband. Our proof is based on general abstract results recently obtained by the authors in [FM], and on a new integral representation formula for the "canonical" solution of the Poisson equation on the ball, that is the unique solution of the equation ∆u = f which is orthogonal to the harmonic functions on the ball. The main technical difficulty of the paper is to establish an asymptotically sharp growth estimate for the kernel of such representation, expressed in terms of its distribution function. We will also consider the situation where the exponential class is endowed with more general Borel measures, and obtain corresponding sharp Moser-Trudinger inequalities of trace type.

Weighted Trudinger - Moser Inequalities and Applications

Bulletin of the South Ural State University. Series "Mathematical Modelling, Programming and Computer Software", 2015

Trudinger Moser inequalities provide continuous embeddings in the borderline cases of the standard Sobolev embeddings, in which the embeddings into Lebesgue L p spaces break down. One is led to consider their natural generalization, which are embeddings into Orlicz spaces with corresponding maximal growth functions which are of exponential type. Some parameters come up in the description of these growth functions. The parameter ranges for which embeddings exist increase by the use of weights in the Sobolev norm, and one is led to consider weighted TM inequalities. Some interesting cases are presented for special weights in dimension two, with applications to mean eld equations of Liouville type.

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.

Weighted Trudinger-Moser inequalities and associated Liouville type equations

Proceedings of the American Mathematical Society

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 us to obtain the existence of a second (mountain-pass) solution for almost all slightly supe...

On Trudinger-Moser type inequalities involving

2008

On Trudinger–Moser type inequalities with logarithmic weights

Journal of Differential Equations, 2015

Trudinger-Moser type inequalities for radial Sobolev spaces with logarithmic weights are considered. The precise Trudinger-Moser growths in dependence on the logarithmic terms, and the corresponding sharp Moser type exponents are determined. In a particular case a critical Trudinger-Moser growth of double exponential type is found.

Sharp Hardy–Sobolev inequalities

Comptes Rendus Mathematique, 2004

Let Ω be a smooth bounded domain in IR N , N ≥ 3. We show that Hardy's inequality involving the distance to the boundary, with best constant (1/4), may still be improved by adding a multiple of the critical Sobolev norm. Résumé Soit Ω un ouvert borné et regulier dans IR N , N ≥ 3. On montre que l'inegalité de Hardy, liéeà la distance au bord, avec meilleure constante (1/4), péutêtre améliorée en ajoutant un multiple de la norme de Sobolev critique.

A Trudinger-Moser inequality in a weighted Sobolev space and applications (original) (raw)

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