Today, we show how fixed point theorems can be used to obtain some existence results for elliptic PDEs. This topic is adapted from Chapter 9 in Evan’s book. The fixed point theorem that we are going to use is the so-called Schaefer Fixed Point Theorem.

Theorem(Schaefer Fixed Point Theorem). Let be a real Banach space. Suppose is a continuous and compact mapping. Assume further that the setis bounded (in ). Then has a fixed point.

Following is the PDE that we are going to demonstrate.

where is bounded with smooth boundary and is a smooth Liptschitz continuous function satisfying the following growth condition

for some constant and all . We claim that

Theorem. If is large enough, there exists a function solving the above PDE.

To prove this result, we do as follows.