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Large viscosity solutions for some fully nonlinear equations

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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.

Mathematics Subject Classification (2000)

35J60 35B40 35B44 35J67 49L25 

Keywords

Boundary blow-up Fully nonlinear operator Keller–Osserman condition Asymptotic behavior Uniqueness 
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© Springer Basel 2013

Authors and Affiliations

  1. 1.Departamento de MatemáticaUniversidad Técnica Federico Santa MaríaValparaisoChile

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