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 .

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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 $u\,|u|$.

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Sir , how did u define the energy functional ? IS it standard ? –  Theorem Jun 11 '12 at 12:24
@Ananda Yes, it is more or less standard. I will add an explanation in the answer. –  Julián Aguirre Jun 11 '12 at 13:00
That would be nice for my better understanding Sir . –  Theorem Jun 11 '12 at 13:17

a steel ring of mean diameter 250mm has a square section 2.5mm by 2.5mm. it is split by narrow radial saw cut. the saw cut is openedd up farthuer by a targential separating force of 0.2N. calculate the extra separation at the saw cut.

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Welcome to MSE! Did you mean to post this as another question? Regards –  Amzoti Apr 16 '13 at 16:49
The intersection of $H^1_0(\Omega)$ and $L^3(\Omega)$ is $H^1_0(\Omega)$ since $\Omega$ is bounded (if it is unbounded, then it is ok) and $H^1_0(\Omega)$ is compactly imbedded in $L^3(\Omega)$. So the correct space $X$ should be $H^1_0(\Omega)$. Am I right? The test functions are also in $H^1_0(\Omega)$. –  daulomb Oct 7 '13 at 23:40