I want to know some reference of Moser's iteration for non homogeneous quasilinear uniformly elliptic PDE, in particular, $u_{ij}(\delta_{ij}+u_iu_j|\nabla u|^2)=0$. I want to know how to deal with the non homogeneous here and whether the Moser's iteration works for non uniformly elliptic PDE. Furthermore, some PDEs like $u_{ji}a_{ij}(x,u,\nabla u)=0$, where $a_{ij}$ is only bounded from below.
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2$\begingroup$ Usually, any technique for analyzing PDE's has to be adapted to a specific situation. You should learn how Moser iteration works in principle and see if you can use interpolation estimates (such as the Gagliardo-Nirenberg inequalities) or whatever else you can find to make it work in this situation. $\endgroup$– Deane YangCommented Dec 16, 2014 at 13:49
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2$\begingroup$ And if you've spent many, many hours (or days) trying without success, you should post a more specific question explaining what you've been able to do so far and pointing out where you got stuck. $\endgroup$– Deane YangCommented Dec 16, 2014 at 13:57
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1$\begingroup$ Just a comment: one interesting quasilinear elliptic PDE which one can apply Moser's iteration to is the minimal surface equation. See e.g. these notes cmouhot.files.wordpress.com/1900/10/mse.pdf $\endgroup$– Otis ChodoshCommented Dec 16, 2014 at 22:50
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