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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Fundamental solutions and two properties of elliptic maximal and minimal operators
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by Patricio L. Felmer and Alexander Quaas PDF
Trans. Amer. Math. Soc. 361 (2009), 5721-5736 Request permission

Abstract:

For a large class of nonlinear second order elliptic differential operators, we define a concept of dimension, upon which we construct a fundamental solution. This allows us to prove two properties associated to these operators, which are generalizations of properties for the Laplacian and Pucci’s operators. If ${\mathcal M}$ denotes such an operator, the first property deals with the possibility of removing singularities of solutions to the equation \[ {\mathcal M}(D^2 u)-u^p=0,\quad \mbox {in}\quad B\setminus \{0\}, \] where $B$ is a ball in $\mathbb {R}^N$. The second property has to do with existence or nonexistence of solutions in $R^N$ to the inequality \[ {\mathcal M}(D^2 u)+u^p\le 0,\quad \mbox {in}\quad \mathbb {R}^N. \] In both cases a common critical exponent defined upon the dimension number is obtained, which plays the role of $N/(N-2)$ for the Laplacian.
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Additional Information
  • Patricio L. Felmer
  • Affiliation: Departamento de Ingeniería Matemática, and Centro de Modelamiento Matemático, UMR2071 CNRS-UChile, Universidad de Chile, Casilla 170 Correo 3, Santiago, Chile
  • Alexander Quaas
  • Affiliation: Departamento de Matemática, Universidad Santa María, Casilla: V-110, Avda. España 1680, Valparaíso, Chile
  • MR Author ID: 686978
  • Received by editor(s): January 4, 2006
  • Received by editor(s) in revised form: May 10, 2007
  • Published electronically: June 16, 2009
  • © Copyright 2009 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 361 (2009), 5721-5736
  • MSC (2000): Primary 35J60; Secondary 35B05, 35B60
  • DOI: https://doi.org/10.1090/S0002-9947-09-04566-8
  • MathSciNet review: 2529911