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 $c$?
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.