Existence of a positive solution to a "semilinear" equation involving Pucci's operator in a convex domain

Abstract

In this article we prove existence of positive solutions for the nonlinear elliptic equation \begin{eqnarray*} \begin{array}{rcll}{{{\mathcal M}_{\lambda,\Lambda}^+}}(D^2u)-\gamma u+f(u) & = & 0 \quad\mbox{in} \quad \Omega ,\\ u & = & 0 \quad \mbox{on}\quad \partial \Omega,\\ \end{array} \end{eqnarray*} where ${{{\mathcal M}_{\lambda,\Lambda}^+}}$ denotes Pucci's extremal operator with parameters $0 s < \lambda\leq \Lambda $ and $\Omega$ is convex smooth domain in ${\hbox{\bf R}}^N$, $N\geq 3$. The result applies to a class of nonlinear functions $f$, including the model cases: i) $\gamma=1$ and $f(s)=s^p$, $1s < p \leq p^+$; and ii) $\gamma=0$, $f(s)=\alpha s+s^p$, $1s <p\leq p^+$, and $ 0\le\alpha s <\mu^+_1$. Here $p^+=\tilde{N}^+/(\tilde{N}^+-2)$, $\tilde{N}^+=\lambda(N-1)/\Lambda+1$, and $\mu^+_1$ is the first eigenvalue of ${{{\mathcal M}_{\lambda,\Lambda}^+}}$ in $\Omega$. Analogous results are obtained for the operator ${{{\mathcal M}_{\lambda,\Lambda}^-}}$.