I am not familiar with the theory of nonlinear PDEs and wonder if there is a theorem that states sufficient and necessary conditions for the existence of weak solutions to the problem
$\begin{cases} -\Delta u= f(u)&\text{ in }\Omega\\ ~~~u=0&\text{ in }\partial \Omega \end{cases}$
Where $f$ is some convex and smooth real-valued function. What particular example can you give?