Removable singularities of some strongly nonlinear elliptic equations |
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Authors: | Juan-Luis Vàzquez Laurent Véron |
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Institution: | 1. Dep. de Ecuaciones Funcionales, Facultad de Ciencias, C.U., Madrid-3, Spain 2. Département de Mathématiques, Faculté des Sciences, Parc de Grandmont, 37200, Tours, France
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Abstract: | Let Ω be some open subset of ?N containing 0 and Ω′=Ω?{0}. If g is a continuous function from ? × ? into ? 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 ?+ and above on ??, then we prove that any subset Σ with 1-capacity zero is a removable singularity for a function u∈L loc ∞ (ω?Σ) satisfying $$\begin{array}{*{20}c} { - div \left( {\frac{{Du}}{{\sqrt {1 + \left| {Du} \right|^2 } }}} \right) + g(.,u) = 0} & {in \mathcal{D}'(\Omega - \Sigma )} \\ \end{array} $$ . |
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