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$\Omega$ is a bounded open subset of $R^n$ $$ -\Delta u -u\ln u=\lambda u \\ u|_{\partial\Omega}=0 $$

What should I read about this eigenvalue question ? I mean some reference or book.I want to answer this eigenvalue question as my homework.

At the beginning, I want to use the fix point theory, but if let $Lu=\frac{-\Delta u -u\ln u}{\lambda}$, I don't know what the $L$ maps suitable space (for example $H_0^2(\Omega)$) to. In fact ,about this question I have asked a question.

Today I read a connected question, but I am unfamiliar with the energy method and don't understand the answer of it. If this is a suitable way for my question, I should read which chapters of Evans' PDE or other books?

I really don't know whether suitable my question is .I just a beginner of PDE. If there are any doubt , please tell me .thanks.

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Enhao Lan
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  • Is this a proper problem? The PDE is not linear, so there will be no eigenspaces or resolvent operators. In what sense do you speak of eigenvalues? – Lutz Lehmann Feb 24 '16 at 10:54
  • @LutzL Why it will be no eigenspaces? – Enhao Lan Feb 24 '16 at 11:59
  • Because the equation is not linear. If $u$ is a solution, in general multiples $c·u$, $c\in \Bbb R$ const., will not be solutions. Not to speak of linear combinations. – Lutz Lehmann Feb 24 '16 at 12:03

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