Fundamental solutions and two properties of elliptic maximal and minimal operators

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.