Morrey Potentials and PDE I

Abstract

In these next two chapters, we use much of our previous theory to examine the regularity of certain elliptic non-linear PDE in domains $\Omega \subset \mathbb{R}^{n}$. So in some sense, we have come full circle, back to the kind of PDEs that originally motivated Morrey in the first place to introduce his Morrey condition. In this chapter, we study the familiar equation: $-\Delta u = u^{p},\mbox{ for }p > n/(n - 2),\;u \geq 0$ on $\Omega;\;\partial \Omega$ smooth. In Chapter 16, we will look at non-linear elliptic systems.