Let $\Omega$ be a smooth bounded domain. Consider the equation $$-\Delta u + (g(x)-\Delta u)^+\varphi(u) = f(x)$$ $$u|_{\partial\Omega} = 0$$ where $f,g$ are smooth functions on $\Omega$ and $\varphi$ is a real valued map which is such that $0 \leq \varphi \leq M$ for some constant $M$.

How do I show existence for this equation?

My problem here is the positive-part term there. I tried to apply a Tikhonov fixed point approach in the spaces $H^2$ to the PDE $$-\Delta u + (g(x)-\Delta w)^+\varphi(u) = f(x)$$ but this requires some weak continuity: if $w_n \rightharpoonup w$ in $H^2$, I would have to show that the corresponding solutions $u_n \to u$ in some space, where $u$ solves the limit problem. But I cannot say anything about the weak convergence of the $(\cdot)^+$ term.

Any existence in Holder or Sobolev spaces would be helpful. Thanks