Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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 .

share|cite|improve this question
up vote 5 down vote accepted

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 $u\,|u|$.

share|cite|improve this answer
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.

share|cite|improve this answer
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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.