I'm reading the first chapter of the book $\textit {a geometric approach to free boundary problems}$ by Caffarelli and Salsa, see the PDF here ftp://nozdr.ru/biblioteka/kolxo3/M_Mathematics/MC_Calculus/MCde_Differential%20equations/Caffarelli%20L.,%20Salsa%20S.%20A%20geometric%20approach%20to%20free%20boundary%20problems%20%28AMS,%202005%29%28ISBN%200821837842%29%28258s%29_MCde_.pdf. The question came from Page 14-15. Let me state my question:

Let $f_{\epsilon}(s)$ be a smooth approximation of Dirac measure (more details can be found in the book), and the support of $f_{\epsilon}(s)$ is the interval $[0,\epsilon]$. Given boundary data $g \in H^1(B_1)$, $g \ge 0$, and now consider the Dirichlet type equation:

$$2 \Delta u=f_{\epsilon}(u),\,u_{\epsilon}(0)=\epsilon,\, u|_{\partial B_1}=g $$

Implicit in the chapter the authors use the existence of smooth solution of the equation above. I've never learned this before, especially there is a constraint on the origin. Can anyone give me some references? Thanks in advance!