Let $\Omega\subset \mathbb R^2$ be a bounded, smooth boundary domain. I am interested in the following operator $$ -\operatorname{div}\left(\frac{\nabla u}{\sqrt{|\nabla u|^2+\epsilon}}\right)=0. $$ where $\epsilon>0$ is a fixed constant, and we apply Neumann boundary condition.
My question: can we somehow have the eigenfunction and eigenvalue of this operator such that the following hold? $$ -\operatorname{div}\left(\frac{\nabla u}{\sqrt{|\nabla u|^2+\epsilon}}\right)=\lambda u, \tag 1 $$ If we do, what would be the regularity of those eigenfunctions? and can they form an Orthonormal basis of $L^2$ as the eigenfunction of standard Laplace operator $-\Delta$ does?
Thank you!
My idea:
First of all, I would like to make sure $(1)$ indeed has a solution. It looks to me we may study the following minimizing problem $$ E(u)=\int_\Omega \sqrt{|\nabla u|^2+\epsilon}dx-\lambda \int_\Omega u^2dx $$ and I am trying to apply mountain pass theorem on it, although $E(0)\neq 0$...
This question is also posted in MO here. I did my research for couple of days but have no luck. So I put it here for more help.