Consider the Schrödinger-Operator with Potential $V\in L^\infty(\Omega)$ with Dirichlet boundary conditions $$ H^D=-\Delta + V $$ and let $u_{i,n}\in H_0^1(\Omega)$ be the first, nonnegative eigenfunctions to eigenvalues $\lambda_{i,n}$ of such an operator on their support. That is $$ \int \nabla u_{i,n} \nabla\phi+ V u_{i,n} \phi - \lambda_{i,n} u_{i,n} \phi\, \mathrm dx =0 $$ for all $\phi \in C_c^\infty(\{u_i>0\})$. In the following situation the $u_{i,n}$ converge against some $u_{i}$ in $H^1$. Furthermore the eigenvalues converge against $\mu_i$ .
Under which circumstances may I follow that $\mu_i$ is an eigenvalue on the support of $u_{i}$ with associated nonnegative eigenfunction $u_i$. This further implies that $\mu_i$ is indeed the first eigenvalue on $\{u_i >0\}$.
Let me drop the index $i$. Denote $\Omega_n=\{u_n>0\}$ and $\Omega_\infty=\{u>0\}$ where $u$ is the limit of the sequence $(u_n)$. If you can establish the following lemma (which states that the domains $\Omega_n$ converge to $\Omega_\infty$ in some sense), then the claim is quite simple. It suffices that $u_n\to u$ weakly.
Lemma: For every compact $K\subset\Omega_\infty$ there is a natural number $N$ so that $K\subset\Omega_n$ for all $n>N$.
Proof of the claim using the lemma: Take any $\phi\in C^\infty_c(\Omega_\infty)$. It follows from the lemma that $\phi\in C^\infty_c(\Omega_n)$ for sufficiently large $n$. If $u_n\to u$ weakly, then it follows from $$ \int \nabla u_n \nabla\phi+ V u_n \phi - \lambda_n u_n \phi\, \mathrm dx =0 $$ that \begin{eqnarray} && \int \nabla u \nabla\phi+ V u \phi - \lambda u \phi\, \mathrm dx \\&=& \lim_{n\to\infty}\int \nabla u_n \nabla\phi+ V u_n \phi - \lambda u_n \phi\, \mathrm dx \\&=& \lim_{n\to\infty}\int \nabla u_n \nabla\phi+ V u_n \phi - \lambda_n u_n \phi\, \mathrm dx + \lim_{n\to\infty}\int (\lambda_n-\lambda) u_n \phi\, \mathrm dx \\&=& 0. \end{eqnarray} To see that the second limit is zero, you can use the fact that weak convergence implies boundedness and estimate $\left|\int(\lambda_n-\lambda)u_n\phi\right|\leq |\lambda_n-\lambda|\|u_n\|_{L^2}\|\phi\|_{L^2}$.