THESIS
2022
Abstract
In this thesis, we study positive solutions to the fractional s order Q-curvature
equation
(−Δ)^{s}u = 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 C
^{1} 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 C
^{1} function K
such that its singular solution u can be arbitrarily large near the origin. When
s ∈ (0, 1) and K is negative in B
_{2}, we construct a sequence of solutions that
blows up in B
_{1}, which is a different phenomenon from the classical Nirenberg
problem....[
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In this thesis, we study positive solutions to the fractional s order Q-curvature
equation
(−Δ)^{s}u = 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 C
^{1} 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 C
^{1} function K
such that its singular solution u can be arbitrarily large near the origin. When
s ∈ (0, 1) and K is negative in B
_{2}, we construct a sequence of solutions that
blows up in B
_{1}, which is a different phenomenon from the classical Nirenberg
problem.
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