Regularity of solutions in semilinear elliptic theory
  We study the semilinear Poisson equation   \begin{equation} \label{pro}   \Delta u = f(x, u) \hskip .2 in \text{in} \hskip .2 in B_1.   \end{equation}   Our main results provide conditions on $f$ which ensure that weak solutions of this equation belong to $C^{1,1}(B_{1/2})$. In some configurations, the conditions are sharp. 
Emanuel Indrei, Andreas Minne, Levon Nurbekyan
math.AP