In this thesis, we study positive solutions to the fractional s order Q-curvature equation
(−Δ)^su = K(x)u^n+2s/n−2s , where s ∈ (0, n/2). When s ∈ N₊ and K ≡ 1, we prove an upper blow up rate and asymptotic symmetry of the singular solutions near the singular set. When s ∈ (1/2, 1), in lower dimensions, we show that for any positive C¹ function K, a singular solution u satisfies an upper blow up rate near the origin. In contrast, when s ∈ (0, 1) or s ∈ N₊, and n > 2s+3, we construct a positive C¹ function K such that its singular solution u can be arbitrarily large near the origin. When s ∈ (0, 1) and K is negative in B₂, we construct a sequence of solutions that blows up in B₁, which is a different phenomenon from the classical Nirenberg problem....[ Read more ]

In this thesis, we study positive solutions to the fractional s order Q-curvature equation
(−Δ)^su = K(x)u^n+2s/n−2s , where s ∈ (0, n/2). When s ∈ N₊ and K ≡ 1, we prove an upper blow up rate and asymptotic symmetry of the singular solutions near the singular set. When s ∈ (1/2, 1), in lower dimensions, we show that for any positive C¹ function K, a singular solution u satisfies an upper blow up rate near the origin. In contrast, when s ∈ (0, 1) or s ∈ N₊, and n > 2s+3, we construct a positive C¹ function K such that its singular solution u can be arbitrarily large near the origin. When s ∈ (0, 1) and K is negative in B₂, we construct a sequence of solutions that blows up in B₁, which is a different phenomenon from the classical Nirenberg problem.