Using Energy method in finding weak solutions of PDE . Let us consider $\Omega \subset R^n$ be open , bounded with smooth boundary. Let $f\in L^2(\Omega)$. How can i use direct energy method to prove that there exists unique weak solution $u\in H_0^1 (\Omega) \cap L^3(\Omega)$ of the following equation. 
$-\triangle u +u|u| =f$ in $\Omega$
$u=0$ in $\partial \Omega$ 
I would like to solve this problem and know the strategies in solving such problems . I would be glad if someone can help. 
Thank you very much . 
 A: Weak solutions of the equation are critical points of the energy functional
$$
E(u)=\int_\Omega\Bigl(\frac12|\nabla u|^2+\frac1{3}|u|^3-f\,u\bigr)dx.
$$
Observe that $E(u)$ is well defined on the Banach space $X=H^1_0(\Omega)\cap L^3(\Omega)$.
To show the existence of a solution it is enough to show that $E$ attains its minimum on $X$. This is done by showing that $E$ is strictly convex and coercive (i.e. $E(u)\to\infty$ if $\|u\|_X\to\infty$).
Uniqueness follows (by a different method) because the function $g(u)=u\,|u|$ is increasing.
There are many books on variational methods. One of them is Introduction à la théorie des points critiques by O. Kavian (in french.) A more general case of your question is solved on page 139.

The following is a formal explanation; I will not go into the details of the correct spaces,...
The weak formulation of the problem is
$$
\int_\Omega\bigl(\nabla u\cdot\nabla u+u\,|u|\,v-f\,v\bigr)dx=0\quad v\text{ a test function.}
$$
The energy functional is a functional $E:X\to\mathbb{R}$ whose derivative with respect to $u$ is given precisely by the weak formulation. What does ths mean? The differential at $u$ is defined as an element $E'(u)$ of de dual of $X$ such that
$$
E(u+v)=E(u)+\langle E'(u),v\rangle+o(\|v\|).
$$
One way to compute it is
$$
\langle E'(u),v\rangle=\lim_{t\to0}\frac{E(u+t\,v)-E(v)}{t}.
$$
Using this it is easy to show that $E(u)$ is in fact the energy functional associated to the equation. Observe that $|u|^3/3$ is a primitive of $u\,|u|$.
