Maximum Principles for Laplacian and Fractional Laplacian with Critical Integrability

IIn this paper, we study themaximum principles for Laplacian and fractional Laplacian with critical integrability. We first consider the critical cases for Laplacian with zeroorder term and first-order term. It is well known that for the Laplacian with zeroorder term - Delta+ c(x) in B-1, c(x). L-p( B-1)(B-1 subset of R-n), the critical case for the maximum principle is p = n/2. We show that the critical condition c(x). L (n/2) ( B-1) is not enough to guarantee the strong maximum principle. For the Laplacian with firstorder term - Delta + b(x)( b(x) is an element of L-p( B-1)), the critical case is p = n. In this case, we establish the maximum principle and strong maximum principle for Laplacian with first-order term. We also extend some of themaximum principles above to the fractional Laplacian. We replace the classical lower semi-continuous condition on solutions for the fractional Laplacian with some integrability condition. Then we establish a series of maximum principles for fractional Laplacian under some integrability condition on the coefficients. These conditions are weaker than the previous regularity conditions. The weakened conditions on the coefficients and the non-locality of the fractional Laplacian bring in some new difficulties. Some new techniques are developed.