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 $0<\lambda_1<\lambda_2\leq\lambda_3<...$, and the corresponding eigenfunction forms a ONB in $L^2$.

My question: for any $\epsilon>0$ fixed, do we have a similar set of eigenfunctions for the following equation? 
$$
-\operatorname{div}\left(\frac{\nabla u}{\sqrt{|\nabla u|^2+\epsilon}}\right)=0.
$$
If there do exists a countable set of eigenfunctions, what about the regularity? Could they form an ONB as the laplace equation?


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.