# American Institute of Mathematical Sciences

May  2008, 7(3): 631-643. doi: 10.3934/cpaa.2008.7.631

## Regularity for positive weak solutions to semi-linear elliptic equations

 1 Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China

Received  July 2007 Revised  November 2007 Published  February 2008

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:

$-\Delta u= f(u)\quad$ in $\Omega,$

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 $f$.

Citation: Li Ma, Lin Zhao. Regularity for positive weak solutions to semi-linear elliptic equations. Communications on Pure & Applied Analysis, 2008, 7 (3) : 631-643. doi: 10.3934/cpaa.2008.7.631
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