I want to continue my previous post on the prescribed Gaussian curvature equations. Still borrowing the idea and technique introduced in the Struwe et al’ paper, today, I want to talk about how one can pass from locally -bounded to pointwise bounded. As always, we are interested in solving the following PDE
For the sake of clarity, let say as and suppose for each , solves the PDE, i.e. the following
holds. As we have already seen, the sequence of solution is -bounded in the region . We now show that such an -boundedness can guarantee that is pointwise bounded from above in . As we shall see later, perhaps, the argument used below only works for the sequence of solutions of the PDE.
To see this, it suffices to prove that
for any but fixed ball . To see this, we first observe from the Trudinger inequality that for each
for some . Note that
Using the mononicity of the exponential function and the fact that is bounded, we can conclude that
is bounded. Hence
is bounded as well. We now let be the unique solution of the auxiliary problem
The standard Schauder estimate tells us that the sequence is bounded in . Hence in view of the Sobolev embedding,
for some constant . Using the equation and the one that solves, we conclude that the function
is harmonic on . It now suffices to show that is bounded from above. To do so, we use the mean value property of harmonics functions. For example, pick any and choose a small ball . Then we can write
Each terms on the right most hand side can be estimated further as follows:
Hence we have just shown that is pointwise bounded from above. It seems that this cannot be true for any sequence of functions rather than those solving the PDE.