Partial regularity for weak solutions of semilinear elliptic equations with supercritical exponents

Abstract

Let Ω; be an open subset in Rn (n ≥ 3). In this paper, we study the partial regularity for stationary positive weak solutions of the equation (1.1) Δu + h1(x)u + h2(x)u α = 0 in Ω. We prove that if α > n+2/n-2, and u ∈ H1 (Ω) ∩ Lα+1(Ω) is a stationary positive weak solution of (1.1), then the Hausdorff dimension of the singular set of u is less than n - 2α+1/α-1, which generalizes the main results in Pacard 1993 and Pacard 1994.