An existence result for a nonlinear transmission problems

Let $Ω^o$ and $Ω^i$ be open bounded subsets of $\mathbb{R}^n$ of class $C^{1,α}$ such that the closure of $Ω^i$ is contained in $Ω^o$. Let $f^o$ be a function in $C^{1,α}(\partialΩ^o)$ and let $F$ and $G$ be continuous functions from $\partialΩ^i\times\mathbb{R}$ to $\mathbb{R}$. By exploiting an argument based on potential theory and on the Leray-Schauder principle we show that under suitable and completely explicit conditions on $F$ and $G$ there exists at least one pair of continuous functions $(u^o, u^i)$ such that $\left\{ \begin{array}{ll} Δu^o=0&\text{in }Ω^o\setminus\mathrm{cl}Ω^i\,,\\ Δu^i=0&\text{in }Ω^i\,,\\ u^o(x)=f^o(x)&\text{for all }x\in\partialΩ^o\,,\\ u^o(x)=F(x,u^i(x))&\text{for all }x\in\partialΩ^i\,,\\ ν_{Ω^i}\cdot\nabla u^o(x)-ν_{Ω^i}\cdot\nabla u^i(x)=G(x,u^i(x))&\text{for all }x\in\partialΩ^i\,, \end{array} \right.$ where the last equality is attained in certain weak sense. In a simple example we show that such a pair of functions $(u^o, u^i)$ is in general neither unique nor local unique. If instead the fourth condition of the problem is obtained by a small nonlinear perturbation of a homogeneous linear condition, then we can prove the existence of at least one classical solution which is in addition locally unique.

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