WEIGHTED TRUDINGER-MOSER INEQUALITIES AND ASSOCIATED LIOUVILLE TYPE EQUATIONS

Autor(es)

Calanchi, Marta ^[1] ; Massa, Eugenio ^[2] ; Ruf, Bernhard ^[1]

Afiliação do(s) autor(es)

^[1] Univ Milan, Dipartimento Matemat, Via Saldini 50, I-20133 Milan - Italy

^[2] Univ Sao Paulo, Inst Ciencias Matemat & Comp, Dept Matemat, Campus Sao Carlos, Caixa Postal 668, BR-13560970 Sao Carlos, SP - Brazil

• Tipo de documento: Artigo Científico
• Fonte: Proceedings of the American Mathematical Society; v. 146, n. 12, p. 5243-5256, DEC 2018.
• Citações Web of Science: 2

Resumo

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. (AU)

• Processo FAPESP: 14/25398-0 - Equações e sistemas elípticos com vários tipos de interação com o espectro
• Beneficiário: Eugenio Tommaso Massa
• Linha de fomento: Auxílio à Pesquisa - Regular