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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.

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    $\begingroup$ what do you mean by "the eigenfunction of the following equation"? Do you just want a solution? Do you want a solution to the equation with the right hand side replaced by $\lambda u$ for some $\lambda$? $\endgroup$ Jul 11, 2016 at 18:34
  • $\begingroup$ @WillieWong The later one. I also update my post. Thx for pointing out! $\endgroup$
    – JumpJump
    Jul 11, 2016 at 18:47
  • $\begingroup$ There is a lot of literature on nonlinear elliptic eigenvalue problems and calculus of variations methods applied to them. Researching it for yourself will be a very valuable experience during which you will acquire skills extremely useful for your future research career. Do not spoil by going to the internet and asking others to do the work for you. $\endgroup$ Jul 12, 2016 at 2:23
  • $\begingroup$ @MichaelRenardy sorry for the trivial question... I did my search for a day but I didn't really find anything useful... Maybe I am heading for the wrong direction. I will continue search on it. Thx for your comment anyway! $\endgroup$
    – JumpJump
    Jul 12, 2016 at 2:34
  • $\begingroup$ @tankonetoone: I did not say that question is trivial. Just that a lot has been done on related problems. $\endgroup$ Jul 12, 2016 at 10:33

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