Suppose that $\Omega\subset \mathbb{R}^n$ is a smooth open region and that $V:\Omega\to \mathbb{R}^+$ be a positive smooth function.

Then we have a family of operators
$$L_\epsilon =-\Delta -\epsilon V.$$
It is straightforward to see that there is an $\epsilon_{crit}(V,\Omega)>0$ so that for $\epsilon<\epsilon_{crit}$ we have
$$
\lambda_1(L_\epsilon)>0
$$
and
$$
\lambda_1(L_{\epsilon_{crit}})=0.
$$

Consider now the following Dirichlet problem $$ L_\epsilon u= 0 \mbox{ where } u|_{\partial \Omega}=1. $$

When $\epsilon<\epsilon_{crit}$ there is a unique solution smooth solution whereas for $\epsilon=\epsilon_{crit}$ there is no smooth solution (which follows from the Hopf boundary maximum principle and Green's identity).

My questions are:

Is there a natural way to make the limit $$ \lim_{\epsilon \nearrow \epsilon_{crit}}u_\epsilon:= u_{\epsilon_{crit}} $$ make sense -- in particular as some sort of singular solution to the Dirichlet problem? Clearly, both the convergence and $ u_{\epsilon_{crit}}$ would need to be singular in an appropriate sense.

If this is the case can one "localize" the singularities for instance restrict where they are allowed to occur in terms of $V$ and $\Omega$?

Has this been considered in the literature at all?