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.　Secondly,　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.　(C)　2018　Elsevier　Inc.　All　rights　reserved.