Regularity for positive weak solutions to semi-linear elliptic equations

Abstract

In this paper, we first derive a new non-linear type inequality for Newtonian potential and then we study the regularity problem for positive weak solutions to the non-linear Laplace equation: -Δu = f(u) in Ω, with f(u) ∈ L1 (Ω). Here Ω is a bounded domain in R n, and f(u) is a regular function with respect to u. We give an apriori estimate for positive weak solutions. We show that under some appropriate assumptions on the non-linear term f, the positive weak solutions are in fact in some local Sobolev space Wloc1,τ (Ω). We also derive a very general local monotonicity formula for variational solutions to the equation above with special nonlinear term f.