Large viscosity solutions for some fully nonlinear equations

Abstract

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.