Hi can anyone complete my solution (or give a better solution /hints to a better solution) to the following problem :
Define : The Sobolev Space $W^{1,4}(\mathbb{T}^{2})$ as $$W^{1,4}(\mathbb{T}^{2})=\{u~:~~ u\in L^4_{loc}(\mathbb{R}^2), ~~ \Delta u\in L^4_{loc}(\mathbb{R}^2),~~ u(x_{1}+1,x_{2})=u(x_{1},x_{2}),~~u(x_{1},x_{2}+1)=u(x_{1}+1,x_{2}),~~ \int_{(0,1)^{2}}u~=0 \} $$
Define : The functional $J:W^{1,4}(\mathbb{T}^2)\to \mathbb{R} $ (for some $f\in L^2(\mathbb{T}^2) $)
$$J(u)=\frac{1}{2}\int_{(0,1)^{2}}|Du|^2 + \int_{(0,1)^{2}}u_{x_{1}}^4+u_{x_{2}}^4 - \int_{(0,1)^{2}}fu$$
Show that if $J(u_{n})\to \inf_{u}J(u)$ then $\{ u_{n}\}$ is bounded in $W^{1,4}(\mathbb{T}^2)$.
$\bf{Hint}$ Use Poincare-Writinger inequality to justify why $\exists C>0$ s.t for any $u\in W^{1,4}(\mathbb{T}^2)$
$$||u||_{W^{1,4}(\mathbb{T}^2)}\leq C|| D u||_{W^{1,4}(\mathbb{T}^2)} $$
Note : $Du=(u_{x_{1}},u_{x_{2}})$
$\bf{My Attempt}$ ( note : we only need to bound the tail of the sequence)
Let $\epsilon>0$ using the definition of limit $\exists N$ s.t $\forall n>N$
$$| J(u_{n})-\inf J(u) | \leq \epsilon $$
Now using definition of infimum it follows
$$J(u_{n})\leq \epsilon +\inf J(u) $$ $$J(u_{n}) \leq \eta ~~~~ \forall n>N $$
Now were done if we can show $||u_{n}||_{W^{1,4}(\mathbb{T}^2)} \leq J(u_{n})$ Im struggling to extract $||u_{n}||_{W^{1,4}}(\mathbb{T}^2)$ from the above because of the $-\int_{(0,1)^2} fu_{n} $ term. I guess its a combination of Holder, $f$ zero average, and the 'hint'.
We have $$ |\int fu| \leq \|f\|_{L^2}\|u\|_{L^2} \leq \varepsilon\|u\|_{L^2}^2+\frac4\varepsilon\|f\|_{L^2}^2 \leq C\varepsilon\|Du\|_{L^2}^2+\frac4\varepsilon\|f\|_{L^2}^2 , $$ where we have used the fact that $\int u=0$ in combination with the Poincare-Wirtinger inequality. Take $\varepsilon>0$ small enough, and get $$ \|Du\|_{L^4}^4\leq\alpha J(u) + \beta $$ for some constants $\alpha,\beta$.