How to asssociate a fuctional to semi linear elliptic boundary value problem

How to asssociate a fuctional to semi linear elliptic boundary value problem

how to associate this functional $$I(u) := \int_{U}\frac{1}{2}\vert Du
\vert^{2} - F(u)\; dx$$ to the Semilinear elliptic boundary-value problem
\begin{cases} -\Delta u=f(u), & \text{in $U$ } \\ u=0, & \text{on
$\partial U$ } \\ \end{cases} where $U$ is bounded subset of
$\mathcal{R}^{n}$ and $F(z):= \int^{z}_{0} f(s) \, ds$ (refer book- Evans
partial differential equation-page number 482, that is application to
semilinear elliptic pde,in section - critical point, of chapter- The
calculus of variation)