Regular perturbation for a elliptic equation?

Question

I focus on the following problem $$u_{xx}+u^3=f+\epsilon g,\quad x\in(0,1),\\u(0)=u(1)=0,\tag{1}$$ If we have known $w$ solves the problem $$w_{xx}+w^3=f,\quad x\in(0,1),\\w(0)=w(1)=0,\tag{2}$$ and for the small perturbation $\epsilon g$ with $0<\epsilon\ll1$, we set $u=w+v_\epsilon$ is the solution of $(1)$. Is there some standard methods to get the existence of $u$ or $v_\epsilon$, and I also need some smallness bound for the solution $v_\epsilon$ in a certain smooth norm (e.g. Sobolev norm)? Any recommendation and reference wil be appreciated！

Answer 1

What you get is that $v/\varepsilon$ is approximately the solution of the linearized equation \begin{align*}v_{xx} + 3\,w^2\,v &= g\\v(0) = v(1) &= 0\end{align*} In particular, you have $$\lVert u - w - \varepsilon \, v\rVert_{H^1} \, \varepsilon^{-1} \to 0$$ as $\varepsilon \to 0$. This can be proven by the implicit function theorem.