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 ofThen 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

where

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 satisfyand

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 problemwith 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 problemwith 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.

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