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

We consider the following nonlinear elliptic equation with singular nonlinearity:Δu−1uα+a1uβ=0inΩ, where α>β>1α>β>1, a>0a>0, and Ω   is an open subset of RnRn, n⩾2n⩾2. Let u∈H1(Ω)u∈H1(Ω) with ∫Ωu1−αdx<∞ and ∫Ωu1−βdx<∞ be a nonnegative stationary solution. If we denote the zero set of u byΣ={x∈Ω:limr→0+1|B(x,r)|∫B(x,r)udxexists, and is equal to0}, we shall prove that the Hausdorff dimension of Σ   is less than or equal to n−2+4α+1.