Let $u\in W^{1,2}_0((a,b))$, $(a,b)=I$ and $\psi'$ the derivative of a convex function $\Psi\in C^1(\mathbb{R})$. Consider $f\in L^2(I)$ and the differential equation $$u''=\psi'(u)+f.$$ I want to intigrate both sides to obtain a weak formulation of this problem, such that I can define what a weak solution is. Therefore I wanted to do this by partial integration of the terms $\int_I u''vdx $and $\int_I \psi'(u)vdx$
$$\int_I u''vdx =\int_I \psi'(u)vdx +\int_I fvdx$$ for $v\in W^{1,2}_0(I)$. But I'm stuck with this $\int_I \psi'(u)vdx$ term, see here antiderivative of $\psi'(u)$ for $u\in W^{1,2}_0((a,b))$ .
How can I get a weak formulation of this differential equation?