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Let me make my question more clear. Suppose I have a nonlinear elliptic PDE, say

$$-\triangle u = u^2$$

and I am solving this problem on a nice domain, say the unit ball $B(0,1)$ and I have the zero boundary condition. I was wondering it is possible to obtain some geometric information of the level set of $u$? For example, it is possible to have a solution that the set, as $c$ is a constant

$$A:=\{x\in B(0,1),\,\,u(x)=c\}$$

is proportion to $B(0,1)$? i.e., $A= \alpha B$ for some constant $\alpha$ dependes on $c$?

spatially
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  • If $\delta u = u^2$ and $u=0$ on the boundary, then isnt $u=0$ a solution to your problem? – 5xum Aug 16 '14 at 22:35
  • If unique continuation holds for your equation, then a solution that is constant on an open set will be identically constant. –  Aug 17 '14 at 04:54
  • @900sit-upsaday: for this question, a solution that is constant on an open set can only be zero on that open set. I somehow suspect that OP is thinking about the super or sub level sets. – Willie Wong Aug 20 '14 at 15:25

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If the solution u is positive, then by a well-known theorem due to Gidas, Ni and Nirenberg, it has to be radially symmetric. Then the level sets are concentric spheres. When u is not positive, in general nothing can be said about the level sets.

Jorge
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