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

Abstract

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 of the fully nonlinear elliptic operator F, 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, 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 appropriate variational formulation. We also obtain an existence result with boundary blow-up in smooth domains.