Metal–organic frameworks (MOFs) attract considerable attention due to tunable structures, tailorable functionalities, and replaceable metal centers, leading to comprehensive applications, such as semiconductors, catalysts, and photocatalysts.

For any n ≥ 3, 0 < m ≤ (n − 2)/n, and constants η > 0, β > 0, α, satisfying α ≤ β(n − 2)/m, we prove the existence of radially symmetric solution of n−1 m ∆v m + αv + βx · ∇v = 0, v > 0, in R n , v(0) = η, without using the phase plane method. When 0 < m < (n − 2)/n, n ≥ 3, and α = 2β/(1 − m) > 0, we prove that the radially symmetric solution v of the above elliptic equation satisfies lim |x|→∞ |x| 2 v(x) 1−m log |x|

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Let n ≥ 3, 0 < m ≤ (n − 2)/n, p > max(1, (1 − m)n/2), and 0 ≤ u 0 ∈ L p loc (R n ) satisfy lim inf R→∞ R −n+ 2 1−m |x|≤R u 0 dx = ∞. We prove the existence of unique global classical solution of u t = n−1 m ∆u m , u > 0, in R n × (0, ∞), u(x, 0) = u 0 (x) in R n . If in addition 0 < m < (n − 2)/n and u 0 (x) ≈ A|x| −q as |x| → ∞ for some constants A > 0, q < n/p, we prove that there exist constants α, β, such that the function v(x, t) = t α u(t β x, t) converges uniformly on every compact subset of R n to the self-similar solution ψ(x, 1) of the equation with ψ(x, 0) = A|x| −q as t → ∞. Note that when m = (n − 2)/(n + 2), n ≥ 3, if g ij = u 4 n+2 δ ij is a metric on R n that evolves by the Yamabe flow ∂g ij /∂t = −Rg ij with u(x, 0) = u 0 (x) in R n where R is the scalar curvature, then u(x, t) is a global solution of the above fast diffusion equation.

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