The purpose of this note is to introduce the squeezing method. As an application, we prove some Liouville type theorem for solutions to logistic equations. Here the text is adapted from a paper by Du and Ma published in J. London Math. Soc. in 2001 [here].
Let prove the following theorem.
Theorem. Let and be a non-negative stationary solution of
Then must be a constant.
The basic ingredients in the proof consist of the following three lemmas. But first of all, let us denote by a (uniformly) second order elliptic operator satisfying the PDE
with smooth, and
for some positive constants , .
Lemma 1 (comparision principle). Suppose is a bounded domain in , and are continuous functions on with and non-negative and not identically zero. Let be positive in and satisfy
where is as above , is continuous and such that is strictly increasing for in the range .
Then in .
Next we obtain
Lemma 2. Let be bounded with smooth boundary and be as above. Suppose that and are smooth positive functions on , and let denote the first eigenvalue of on under Dirichlet boundary condition on . Then the problem
with zero Dirichlet boundary
has a unique positive solution for every , and the unique solution satisfies
uniformly on any compact subset of as .
Lastly, we have
Lemma 3. As in Lemma 2, the following problem
with infinite Dirichlet boundary
has a unique positive solution for each , and the unique solution satisfies
uniformly on any compact subset of as . Here by the infinite boundary condition we mean as .
We are now in a positive to prove the Liouville type theorem.
Due to the Harnack inequality, we shall focus on positive solutions. Assuming is an entire solution where , we prove that .
Indeed, let be an arbitrary point in . For any , let us define
It is clear to see that verifies
Let denote the unit ball with center . From Lemma 2, for all sufficiently large, there is some solving
with property that
at as . Using Lemma 1, we conclude that in and hence
Let be the unique solution to the following
Using Lemma 1 and Lemma 3 we can show that
Letting we see that
As be arbitrary, we conclude that . For the case , we refer the reader to the original paper.