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Weighted Trudinger-Moser inequalities and associated Liouville type equations


Authors: Marta Calanchi, Eugenio Massa and Bernhard Ruf
Journal: Proc. Amer. Math. Soc. 146 (2018), 5243-5256
MSC (2010): Primary 35J25, 35B33, 46E35
DOI: https://doi.org/10.1090/proc/14189
Published electronically: August 10, 2018
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Abstract: 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 supercritical parameters using the Struwe monotonicity trick. This result is in contrast to the non-weighted case, where positive solutions do not exist (in star-shaped domains) in the critical and supercritical cases.


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Additional Information

Marta Calanchi
Affiliation: Dipartimento di Matematica, Universitá di Milano, Via Saldini 50, 20133 Milano, Italia
Email: marta.calanchi@unimi.it

Eugenio Massa
Affiliation: Departamento de Matemática, Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo - Campus de São Carlos, Caixa Postal 668, 13560-970, São Carlos SP, Brazil
Email: eug.massa@gmail.com

Bernhard Ruf
Affiliation: Dipartimento di Matematica, Universitá di Milano, Via Saldini 50, 20133 Milano, Italia
Email: bernhard.ruf@unimi.it

DOI: https://doi.org/10.1090/proc/14189
Keywords: Trudinger-Moser inequality, Liouville type equations.
Received by editor(s): November 22, 2017
Received by editor(s) in revised form: March 29, 2018
Published electronically: August 10, 2018
Additional Notes: The first and third authors were partially supported by INdAM-GNAMPA Project 2016.
The second author was supported by grant #2014/25398-0, São Paulo Research Foundation (FAPESP) and grants #308354/2014-1, #303447/2017-6, CNPq/Brazil.
Communicated by: Joachim Krieger
Article copyright: © Copyright 2018 American Mathematical Society

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