Super-linear elliptic equation for the Pucci operator without growth restrictions for the data

Details Super-linear elliptic equation for the Pucci operator without growth restrictions for the data Super-linear elliptic equation for the Pucci operator without growth restrictions for the data

2007-12-09

arxiv.org/abs/0712.1331v1

Mathematics

Analysis of PDEs

Scientific paper

In this paper we deal with existence and uniqueness of solution to super-linear problems for the Pucci operator: $$ -\M^+(D^2u)+|u|^{s-1}u=f(x) \quad {in} \RR^n, $$ where $s>1$ and $f$ satisfies only local integrability conditions. This result is well known when, instead of the Pucci operator, the Laplacian or a divergence form operator is considered. Our existence results use the Alexandroff-Bakelman-Pucci inequality since we cannot use any variational formulation. For radially symmetric $f$ we can prove our results under less local integrability assumptions, taking advantage of an appropriate variational formulation. We also obtain an existence result with boundary explosion in smooth domains.

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