# A constrained variational problem

Let $\Omega \subset \mathbb{R}^3$ be open, bounded and non-empty and consider the functional
$\mathcal{F}(u) = \int_\Omega |\nabla u|^4 - fu dx$
for $u \in W^{1,4}_0(\Omega)$ where $f \in L^{\frac{4}{3}}(\Omega)$ is fixed. Prove that there exists a minimizer of $\mathcal{F}$ satisfying the constraint $\int_\Omega g(u) dx =0$ where $g \in C(\mathbb{R})$ and $g(0)=0$.

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Erdos
4.7K
• why would W^{1,4} be embedded in C^{3,1/4}?

• and further Omega is not compact

• I added some notes at the end of the proof.

• This took me a long time to answer. Please offer higher bounties in the future.

• thanks for the note, but doesn't k need to be greater than r?

• I revised my solution, we should indeed have r=0, but the argument works regardless.

• thanks, i also thinks r=0 is sufficient