The problem is as follows: $u_k: \Omega_k \rightarrow \mathbb{R}$ such that $$-div(A(x,\frac{x}{k})\nabla(u_k)) = f(u_k)$$ in $\Omega_k$ $$-A(x,\frac{x}{k})\nabla(u_k)\nu = -kg$$ on $\Gamma_k$ $$u_k=0$$ on the boundary of $\Omega$
For now, we can assume that $\Omega_k$ is a domain that is dependent on a factor $k$, $0<k<1$. We take $A$ to be elliptic and in $L^\inf$, $g,f$ are both continuous.
I have a theorem that says that if $f$ is Lipschitz-continuous for $L>0$ then, a solution of this problem will be unique (if $L$ is small enough).
I am not sure how to begin to show this. I am familiar with the Picard proof that uses Banach fixed point theorem, or just regular $w=u_1-u_2$ type of proofs, but I am not able to apply those ideas here.