## Fundamental solutions and two properties of elliptic maximal and minimal operators

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- by Patricio L. Felmer and Alexander Quaas PDF
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**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.## References

- Alain Bensoussan and Jacques-Louis Lions,
*Applications of variational inequalities in stochastic control*, Studies in Mathematics and its Applications, vol. 12, North-Holland Publishing Co., Amsterdam-New York, 1982. Translated from the French. MR**653144** - Lipman Bers,
*Isolated singularities of minimal surfaces*, Ann. of Math. (2)**53**(1951), 364–386. MR**43335**, DOI 10.2307/1969547 - Isabeau Birindelli and Françoise Demengel,
*Comparison principle and Liouville type results for singular fully nonlinear operators*, Ann. Fac. Sci. Toulouse Math. (6)**13**(2004), no. 2, 261–287 (English, with English and French summaries). MR**2126744** - Haïm Brézis and Laurent Véron,
*Removable singularities for some nonlinear elliptic equations*, Arch. Rational Mech. Anal.**75**(1980/81), no. 1, 1–6. MR**592099**, DOI 10.1007/BF00284616 - Luis A. Caffarelli and Xavier Cabré,
*Fully nonlinear elliptic equations*, American Mathematical Society Colloquium Publications, vol. 43, American Mathematical Society, Providence, RI, 1995. MR**1351007**, DOI 10.1090/coll/043 - Luis A. Caffarelli, Basilis Gidas, and Joel Spruck,
*Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth*, Comm. Pure Appl. Math.**42**(1989), no. 3, 271–297. MR**982351**, DOI 10.1002/cpa.3160420304 - Wen Xiong Chen and Congming Li,
*Classification of solutions of some nonlinear elliptic equations*, Duke Math. J.**63**(1991), no. 3, 615–622. MR**1121147**, DOI 10.1215/S0012-7094-91-06325-8 - C. B. Clemons and C. K. R. T. Jones,
*A geometric proof of the Kwong-McLeod uniqueness result*, SIAM J. Math. Anal.**24**(1993), no. 2, 436–443. MR**1205535**, DOI 10.1137/0524027 - Alessandra Cutrì and Fabiana Leoni,
*On the Liouville property for fully nonlinear equations*, Ann. Inst. H. Poincaré C Anal. Non Linéaire**17**(2000), no. 2, 219–245 (English, with English and French summaries). MR**1753094**, DOI 10.1016/S0294-1449(00)00109-8 - Ennio De Giorgi and Guido Stampacchia,
*Sulle singolarità eliminabili delle ipersuperficie minimali*, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. (8)**38**(1965), 352–357 (Italian). MR**187158** - Patricio L. Felmer and Alexander Quaas,
*On critical exponents for the Pucci’s extremal operators*, Ann. Inst. H. Poincaré C Anal. Non Linéaire**20**(2003), no. 5, 843–865 (English, with English and French summaries). MR**1995504**, DOI 10.1016/S0294-1449(03)00011-8 - Patricio L. Felmer and Alexander Quaas,
*Critical exponents for uniformly elliptic extremal operators*, Indiana Univ. Math. J.**55**(2006), no. 2, 593–629. MR**2225447**, DOI 10.1512/iumj.2006.55.2864 - Basilis Gidas,
*Symmetry properties and isolated singularities of positive solutions of nonlinear elliptic equations*, Nonlinear partial differential equations in engineering and applied science (Proc. Conf., Univ. Rhode Island, Kingston, R.I., 1979) Lecture Notes in Pure and Appl. Math., vol. 54, Dekker, New York, 1980, pp. 255–273. MR**577096** - B. Gidas and J. Spruck,
*Global and local behavior of positive solutions of nonlinear elliptic equations*, Comm. Pure Appl. Math.**34**(1981), no. 4, 525–598. MR**615628**, DOI 10.1002/cpa.3160340406 - David Gilbarg and Neil S. Trudinger,
*Elliptic partial differential equations of second order*, 2nd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. MR**737190**, DOI 10.1007/978-3-642-61798-0 - H. Ishii and P.-L. Lions,
*Viscosity solutions of fully nonlinear second-order elliptic partial differential equations*, J. Differential Equations**83**(1990), no. 1, 26–78. MR**1031377**, DOI 10.1016/0022-0396(90)90068-Z - Robert Jensen,
*The maximum principle for viscosity solutions of fully nonlinear second order partial differential equations*, Arch. Rational Mech. Anal.**101**(1988), no. 1, 1–27. MR**920674**, DOI 10.1007/BF00281780 - Denis A. Labutin,
*Removable singularities for fully nonlinear elliptic equations*, Arch. Ration. Mech. Anal.**155**(2000), no. 3, 201–214. MR**1808368**, DOI 10.1007/s002050000108 - Denis A. Labutin,
*Isolated singularities for fully nonlinear elliptic equations*, J. Differential Equations**177**(2001), no. 1, 49–76. MR**1867613**, DOI 10.1006/jdeq.2001.3998 - Carlo Pucci,
*Maximum and minimum first eigenvalues for a class of elliptic operators*, Proc. Amer. Math. Soc.**17**(1966), 788–795. MR**199576**, DOI 10.1090/S0002-9939-1966-0199576-1 - Carlo Pucci,
*Operatori ellittici estremanti*, Ann. Mat. Pura Appl. (4)**72**(1966), 141–170 (Italian, with English summary). MR**208150**, DOI 10.1007/BF02414332 - James Serrin,
*Removable singularities of solutions of elliptic equations*, Arch. Rational Mech. Anal.**17**(1964), 67–78. MR**170095**, DOI 10.1007/BF00283867 - James Serrin,
*Removable singularities of solutions of elliptic equations. II*, Arch. Rational Mech. Anal.**20**(1965), 163–169. MR**186919**, DOI 10.1007/BF00276442 - James Serrin and Henghui Zou,
*Cauchy-Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities*, Acta Math.**189**(2002), no. 1, 79–142. MR**1946918**, DOI 10.1007/BF02392645 - Laurent Véron,
*Singularities of solutions of second order quasilinear equations*, Pitman Research Notes in Mathematics Series, vol. 353, Longman, Harlow, 1996. MR**1424468**

## 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