  We show that all nonnegative solutions of the critical semilinear elliptic
equation involving the regional fractional Laplacian are locally universally
bounded. This strongly contrasts with the standard fractional Laplacian case.
Second, we consider the fractional critical elliptic equations with nonnegative
potentials. We prove compactness of solutions provided the potentials only have
non-degenerate zeros. Corresponding to Schoen's Weyl tensor vanishing
conjecture for the Yamabe equation on manifolds, we establish a Laplacian
vanishing rate of the potentials at blow-up points of solutions.
