Let $\Omega,\Omega^*$ be disks in $\mathbb{R}^2$, such that $\Omega^*\subsetneq\Omega$ and their boundaries meet at one point (so they are tangent at that point; consider $N((1,0),1)$ and $N((2,0),2)$ for example)
Would there be $u\in C^{\infty}(\Omega)$ s.t. $u|_{\partial\Omega}=u|_{\partial\Omega^*}=0$, $\Delta u>0$ on $int(\Omega\setminus\Omega^*)$, $\Delta u<0$ on $int(\Omega^*)$ and $\Delta u=0$ on $\partial\Omega\cup\partial\Omega^*$?