Superlinear elliptic equation for fully nonlinear operators without growth restrictions for the data

Quaas, Alexander; Felmer, Patricio; Esteban, Maria J.

Quaas, Alexander; Felmer, Patricio; Esteban, Maria J. (2010), Superlinear elliptic equation for fully nonlinear operators without growth restrictions for the data, Proceedings of the Edinburgh Mathematical Society, 53, 1, p. 125-141. http://dx.doi.org/10.1017/S0013091507001393

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Type

Article accepté pour publication ou publié

External document link

https://hal.archives-ouvertes.fr/hal-00195081

Date

2010

Journal name

Proceedings of the Edinburgh Mathematical Society

Volume

Number

Publisher

Scottish Academic Press

Pages

125-141

Abstract (EN)

We deal with existence and uniqueness of the solution to the fully nonlinear equation−F(D2u)+ |u|s−1u = f(x)in Rn,where s> 1 and f satisﬁes only local integrability conditions. This result is well known when, instead ofthe fully nonlinear elliptic operator F, the Laplacian or a divergence-form operator is considered. Ourexistence results use the Alexandroﬀ–Bakelman–Pucci inequality since we cannot use any variationalformulation. For radially symmetric f, and in the particular case where F is a maximal Pucci operator,we can prove our results under fewer integrability assumptions, taking advantage of an appropriatevariational formulation. We also obtain an existence result with boundary blow-up in smooth domains.

Subjects / Keywords

Pucci operator; super-linear elliptic problem; local data.; boundary explosion