The Euler-Lagrange equation associated with $L(p,z,x)=\frac{1}{2}(|p|^2+f(x)z)$

Question

I'm studying the calculus of variations by referring to the PDE book by Lawrence C. Evans (chapter 8, 2nd edition). After learning the Euler-Lagrange equation $$-\sum_{i=1}^n(L_{p_i}(Du,u,x))_{x_i}+L_z(Du,u,x)=0$$ associated with the energy functional $$I[w]=\int L(Dw(x),w(x),x),$$ I got myself an exercise from some other book, which is to minimize the functional $$A(v)=\int\frac{1}{2}(|\nabla v|^2+fv).\tag{$*$}$$ I don't bother about the minimizing stuff; all I want to do is find the correct form of its Euler-Lagrange equation. And for some reasons, the author implicitly suggests that the Euler-Lagrange equation is the Poisson equation $$\Delta v=f,\tag{$**$}$$ which begs a question: is there any typo in $(*)$? Based on what I just learned, we have $$L(p,z,x)=\frac{1}{2}(|p|^2+f(x)z)$$ if $(*)$ is to be considered. Then some calculus gives us $$\Delta v=\frac{1}{2}f,$$ a result differing from the expectation $(**)$ by a factor $1/2$. So my guess would be that something is wrong with the functional $A$. Should we have instead $$A(v)=\int(\frac{1}{2}|\nabla v|^2+fv)?$$ I was trying to contact the author, but it didn't do much good. Much is appreciated if someone could give me some confidence. Thank you so much.