Local uniqueness of the solutions for a singularly perturbed nonlinear nonautonomous transmission problem

We consider the Laplace equation in a domain of ${\mathbb{R}}^n$, $n\ge 3$, with a small inclusion of size $\epsilon$. On the boundary of the inclusion we define a nonlinear nonautonomous transmission condition. For $\epsilon$ small enough one can prove that the problem has solutions. In this paper, we study the local uniqueness of such solutions.