Weighted Trudinger - Moser Inequalities and Applications (original) (raw)

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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.

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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...

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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.

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

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