does anyone know if there's a theory for the following problem: Optimize the task
$\begin{align*} T_\phi(\tilde{u})&=\inf\limits_u T_\phi(u)\\ Au&=b\\ u&\in L^p(\Omega),\,\Omega\subset \mathbb{R}^n \end{align*}$
for (nonlinear map) $A:L^p(\Omega)\to Z$, with $Z$ arbitrary local convex topological vector space and $T_\phi:L^p(\Omega)\to \mathbb{R}\cup\{\pm\infty\}$ defined by
$\int\limits_{\Omega}\phi(u(x))\,dx$
a weakly lower semiconinous convex function, with weakly compact level set. The map $\phi$ is also lower semiconinous convex.
Or exist there a theorem, that such a problem possess a optimal solution?