The following are some of the discussion 10 years ago, or earlier.
Problem 1 Consider harmonic function $ - \ Delta u = 0 \ \ \ in \ \ R ^ n $, $ n \ geq2 $, and $ u (x) \ geq - (1+ | x |) ^ {\ alpha } $, where $ \ alpha \ in (0,1) $, demonstrated: $ u $ will be constant.
Proof: (1) considering direct $ u- \ inf_ {B_ {2R}} u $ Harnack inequality used in the $ B_R $, then
$$ \ sup_ B_ {R} {U} \ {B_ inf_ {2R}} u \ leq C (u (0) - \ inf_ B_ {{u}} 2R). $$
Then
$$\sup_{B_{R}}u\leq C|u(0)|+(1-C)\inf_{B_{2R}}u\leq \leq C|u(0)|+(C-1)(1+2R)^{\alpha},$$
So there
$$\sup_{B_{R}}|u|\leq C|u(0)|+C(1+2R)^{\alpha}$$
Finally, the estimated gradient can be obtained harmonic functions conclusions.
(2) The second method is a method using the average gradient estimation equation derived, and combining the integral theorem to know $ \ nabla u \ equiv 0 $. Oleinik see the "Lecture PDE" specific details of this course order problem also can be extended to control.
Problem 2 . Consider a bounded plane down and the full functions of the two-dimensional case, it certainly is a constant. details as follows:
$$u\in C^2(R^2),\ \ \ -\Delta u\leq0\ \ in \ \ R^2, \sup\limits_{R^2}u=0, \ \ then\ \ u\equiv 0. $$
Proof: The first scenario: If $ u (0) = 0 $, the strong maximum principle known by the conclusion established.
Second case: if $ u (0) = - m <0 $, the following proved impossible.
The apparent continuity exists $ \ delta> 0 $ such that, $ \ forall \ | x | \ leq \ delta, \ \ u (x) \ leq - \ frac {m} {2} <0 $, then the outside consider the use of solutions of basic configuration of a gate function. $ For any \ epsilon> 0 $, take
$$v_{\epsilon}(x)=-\frac{m}{2}+\epsilon \ln(\frac{|x|}{\delta}),$$
Theorem readily known by the comparison $ u \ leq v _ {\ epsilon} $ in $ R ^ 2 \ {x: | x |> \ delta \}. $
Finally, make $ \ epsilon \ rightarrow0 + $ known,
$$u\leq -\frac{m}{2} \mbox{ in} \{x:|x|>\delta\}.$$
Thus there will be $ u \ leq - \ frac {m} {2} $ in $ R ^ 2 $, which is assumed $ \ sup \ limits_ {R ^ 2} u = 0 $ contradict. QED
Question 3 Consider harmonic function $ - \ Delta u = 0 \ \ \ in \ \ R ^ n $, $ n \ geq2 $, and $ u (x) \ in L ^ p (R ^ n) $, $ p > 0 $, then $ u \ equiv 0 $.
Proof: $ p \ when geq 1 $, just use the mean value formula and Holder inequality, as interpolation or other circumstances may be considered directly using the local maximum principle Moser iteration. QED
Question 4 Can singularity problem on bounded to reconcile. Capacity is introduced to describe and intuitive to use Haussdorff measure determination.
Question 5 . Bocher theorem for bounded reconcile outlier of singularities.
4,5 question is worth studying, they can be extended to other elliptic, parabolic equation up.