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

Regularity for positive weak solutions to semi-linear elliptic equations

Li Ma 1, and  Lin Zhao 1,

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 $f$.

Keywords: monotonicity formula., weak solutions, regularity, Newtonian potential.
Mathematics Subject Classification: 35Jx.
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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