Let $\Omega$ be a regular bounded open subset of $\mathbb{R}^3$. The problem is to solve the following pde:
$\left\{\begin{array}{c c}-\Delta u = u^3 & (\Omega)\\u = 0 &(\partial\Omega)\end{array}\right.$
Here are the questions (this is not homework but last year's exam):
- Prove the existence of a solution for (ie. a $v$ such that it achieves the minimum): $\inf \;\left\{ \int_\Omega |\nabla v|^2 : v\in H^1_0(\Omega), \int v^4 =1 \right\}$
- Prove that if $v$ solves $(1)$ then there is a $\lambda > 0$ such that $-\Delta v = \lambda v^3$.
- Conclude.
I was able to prove 1 by considering a sequence $(v_n)$ such that $\|\nabla v\|^2$ converges towards the $\inf$, obtain a weak limit by Banach-Alaoglu, Rellich-Kondrakov to get a limit in $L^2$ and Fatou's lemma to conclude. And if I were able to prove 2, I would have an answer for 3 by simply scaling $v$. But I can't figure out how to prove 2. It sounds an awful lot like Stampacchia's theorem, but unfortunately the unit sphere of $L^4$ isn't convex, and even if it were I don't see how the $\lambda$ would appear or how it would relate to $v^3$; besides Stampacchia's theorem applies only for bilinear forms. How should I go about proving this?