Let $\alpha \geq 0$ be a parameter, and suppose for each $\alpha$, we have two functions $v_{\alpha}, w_{\alpha} \in C^2(\mathbb{R}^2)$ (uniformly bounded) such that $w_{\alpha}>v_{\alpha}$ in $\Omega_{\alpha}$, $w_{\alpha}=v_{\alpha}$ on $\partial \Omega_{\alpha}$, and $$\Delta (w_{\alpha}-v_{\alpha})+e^{w_{\alpha}}-e^{v_{\alpha}}=0 \ \ \ \ \text{on} \ \ \ \ \Omega_{\alpha},$$ or equivalently $w_{\alpha}-v_{\alpha}$ is a positive solution of the linear elliptic PDE $$\Delta (w_{\alpha}-v_{\alpha})+\gamma_{\alpha}(w_{\alpha}-v_{\alpha})=0 \ \ \ \ \text{on} \ \ \ \ \Omega_{\alpha},$$ where $\gamma_{\alpha}=\frac{e^{w_{\alpha}}-e^{v_{\alpha}}}{w_{\alpha}-v_{\alpha}}$ and $\Omega_{\alpha} \subset \mathbb{R}^2$.
Suppose $\Omega_{\alpha}, v_{\alpha}, w_{\alpha}$ are continuous with respect to the parameter $\alpha$. Also assume that for every connected component $\omega$ of $\Omega_{\alpha}$ we have $|\omega|>C$ for some constant $C>0$.
Suppose $\Omega_0$ is connected. I wonder whether it is possible for $\Omega_{\alpha}$ to become disconnected for some $\alpha_1 >0$. i.e. $\Omega_{\alpha_1}= \Omega_1 \cup \Omega_2$ where $\overline{\Omega_1} \cap \overline{\Omega_2}=\emptyset$ with $\Omega_i \neq \emptyset$, $i=1,2$. Basically I wonder if the connected open region $\Omega_{\alpha}$ can continuously break up into two disjoint regions with areas at least $C>0$.
This problem comes up in the study of symmetry of solutions of elliptic PDE with exponential nonlinearity.