Maximum principles for Laplacian and fractional Laplacian with critical integrability

Abstract

In this paper, we study maximum principles for Laplacian and fractional Laplacian with critical integrability. We first consider $-\Delta u(x)+c(x)u(x)\geq 0$ in $B_1$ where $c(x)\in L^{p}(B_1)$, $B_1\subset \mathbf{R}^n$. As is known that $p=\frac{n}{2}$ is the critical case. We show that the maximum principle holds for $p\geq \frac{n}{2}$. On the other hand, the strong maximum principle requires $p>\frac{n}{2}$. In fact, we give a counterexample to illustrate that no matter how small $\|c\|_{L^{p}(B_1)}$ is, the strong maximum principle is false as $p=\frac{n}{2}$. Next, we investigate $-\Delta u(x)+\vec{b}(x)\cdot \nabla u(x)\geq 0$ in $B_1$ where $\vec{b}(x)\in L^p(B_1)$. Here $p=n$ is the critical case. In contrast to the previous case, the maximum principle and strong maximum principle both hold for $p\geq n$. We also extend some of the results above to fractional Laplacian. The non-locality of the fractional Laplacian brings in some new difficulties. Some new methods are needed