Under appropriate conditions on f .x; t/, we prove the existence of viscosity solutions to 1 u D f .x; u/ that take prescribed continuous data on the boundary of bounded domains. As an application, singular boundary value problems are investigated. These problems are shown to admit viscosity solutions and their asymptotic behavior near the boundary is analyzed. Maximum and comparison principles are used as the main tools in these investigations.

We study the existence of positive viscosity solutions to Trudinger's equation for cylindrical domains Ω × [0, T ), where Ω ⊂ IR n , n ≥ 2, is a bounded domain, T > 0 and 2 ≤ p < ∞. We show existence for general domains Ω, when n < p < ∞. For 2 ≤ p ≤ n, we prove existence for domains Ω that satisfy a uniform outer ball condition. We achieve this by constructing suitable sub-solutions and super-solutions and applying Perron's method.

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In this work we study non-negative singular infinity-harmonic functions in the half-space. We assume that solutions blow-up at the origin while vanishing at infinity and on a hyperplane. We show that blow-up rate is of the order |x| −1/3 .

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