Large viscosity solutions for some fully nonlinear equations

We study existence, uniqueness and asymptotic behavior near the boundary of solutions of the problem $$left{begin{array}{ll}-F(D^{2} u) + beta (u) = f quad {rm in} , Omega, u = + infty quad quad quad quad quad quad ,,,, {rm on}, partial Omega, end{array} right.quad quad quad quad quad {rm (P)}$$ where Ω is a bounded smooth domain in ${{mathbb R}^N, N >1 , F}$ is a fully nonlinear elliptic operator and β is a nondecreasing continuous function. Assuming that β satisfies the Keller–Osserman condition, we obtain existence results which apply to ${f in L^infty_{loc}(Omega)}$ or f having only local integrability properties where viscosity solutions are well defined, i.e. ${f in L^N_{loc}(Omega)}$ . Besides, we find the asymptotic behavior near the boundary of solutions of (P) for a wide class of functions ${f in mathcal{C}(Omega)}$ . Based in this behavior, we also prove uniqueness.