Partial regularity for solutions of a nonlinear elliptic equation with singular nonlinearity

We consider the following nonlinear elliptic equation with singular nonlinearity: u − 1 uα +a 1 uβ =0 inΩ, where α >β >1, a >0, and Ω is an open subset of Rn, n 2. Let u ∈ H1(Ω) with Ω u1−α dx <∞and Ω u1−β dx <∞be a nonnegative stationary solution. If we denote the zero set of u by Σ = x ∈ Ω: lim r→0+ 1 |B(x, r)| B(x,r) udx exists, and is equal to 0 , we shall prove that the Hausdorff dimension of Σ is less than or equal to n−2+ 4 α+1 . © 2007 Elsevier Inc. All rights reserved