Let $\Omega\subset \mathbb R^2$ be open bounded with smooth boundary. It is well known that the laplace equation
$$
-\Delta u=0
$$
has a set of eigenvalues $0<\lambda_1<\lambda_2\leq\lambda_3<...$.

My question, for any $\epsilon>0$ fixed, do we have the eigenfunction for the following equation?
$$
-\operatorname{div}\left(\frac{\nabla u}{\sqrt{|\nabla u|^2+\epsilon}}\right)=0.
$$
I think, without $\epsilon$ term the above equation is ill-posted. But how about adding $\epsilon$ term? would it be helpful?

Thank you!

PS: by eigenfunction I am looking for solution like
$$
-\operatorname{div}\left(\frac{\nabla u}{\sqrt{|\nabla u|^2+\epsilon}}\right)=\lambda u,
$$
and also I am interested in regularity as well.