I am using finite element software to solve ODEs and PDEs. It requires the semilinear form. E.g When solving the poisson equation $$\nabla\cdot((1+u^2)\nabla u) = -f$$ the semilinear form $$\int_\Omega (1+u^2)\nabla u\cdot\nabla v - fvdx$$ is required.
What I would like to know is the semilinear form for the equation $$\nabla^2 u = -f(u)$$
Assuming that $\nabla^2u$ is the Laplacian, it would most likely be
$$\int_\Omega\nabla u\cdot\nabla v + f(u)v.$$
Essentially it should be the nonlinear form that defines the weak formulation?