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: <p align="center"> $-\Delta u= f(u)\quad$ in $\Omega,$ <p align="left" class="times"> with $f(u)\in L^1(\Omega)$. Here $\Omega$ 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 $W_{l o c}^{1,\tau}(\Omega)$. We also derive a very general local monotonicity formula for variational solutions to the equation above with special nonlinear term $f$.