Let $\Omega \subset \mathbb{R}^n$ is a bounded domain with smooth boundary.Prove there exists a positive constant $\epsilon_0$ so that for all real numbers $\epsilon<\epsilon_0$ and $f\in L^2(\Omega)$, there exists a unique $u\in H_0^1(\Omega)$ so that
\begin{equation}
-\Delta u+\epsilon \sin u=f
\end{equation}
in the sense of distribution.
For the uniqueness part we may use energy method, just subtract two equations of different weak solutions and then multiply both sides $\varphi_\epsilon *u$.I am not sure if it works.
I have no idea about the existence part, I know some functional analysis methods such as Riesz representation theorem can deal with linear equations, but how to solve nonlinear equations? Is there any good book given an introduction to nonlinear equations?(I'm familiar with real analysis and functional analysis)
Best Answer
In the weak formulation, you can rewrite the problem in $X=H_0^1(\Omega)$ in the form $$J(u)+\varepsilon G(u)=F$$ where $J,G\colon X\to X^*\cong X$ and $F\in X^*\cong X$ are defined by $\langle J(u),v\rangle=-\int_\Omega\nabla u\nabla v$, $\langle G(u),v\rangle=\int_\Omega\sin u v$, and $\langle F,v\rangle=\int_\Omega fv$, respectively.
Now one can show:
After rewriting the equation in the form $u=J^{-1}(-\varepsilon G(u)+F)$ you have now a fixed point problem, and for $\varepsilon<\lVert J^{-1}\rVert/L$ with $L$ being the Lipschitz constant of $G$, the operator on the right-hand side is a contraction. The existence and uniqueness follows from Banach–Caccioppoli.
If you are interested only in the existence part, observe that $G$ and thus the operator on the right-hand side are compact, continuous, and bounded, and you can apply Schauder's fixed point theorem.
The field of nonlinear analysis is wide. Probably one of the best (but very comprehensive) overviews is the monograph of Deimling "Nonlinear Analysis".