Removable singularities of some strongly nonlinear elliptic equations

Abstract

Let Ω be some open subset of ℝ^N containing 0 and Ω′=Ω−{0}. If g is a continuous function from ℝ × ℝ into ℝ satisfying some power like growth assumption, then any u∈L ^∞_loc (Ω′) satisfying

$$\begin{array}{*{20}c} { - div (Du \left| {Du} \right|^{p - 2} ) + g(.,u) = 0} & {in \mathcal{D}'(\Omega ')} \\ \end{array} $$

, remains bounded in Ω and satisfies the equation in D'(Ω). We give extensions when the singular set is some compact submanifold of Ω. When g is bounded below on ℝ⁺ and above on ℝ⁻, then we prove that any subset Σ with 1-capacity zero is a removable singularity for a function u∈L ^∞_loc (ω−Σ) satisfying

$$\begin{array}{*{20}c} { - div \left( {\frac{{Du}}{{\sqrt {1 + \left| {Du} \right|^2 } }}} \right) + g(.,u) = 0} & {in \mathcal{D}'(\Omega - \Sigma )} \\ \end{array} $$

.