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.