Q:

Regular perturbation for a elliptic equation?

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！

A:

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.

