The extremal solution of a boundary reaction problem

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Positive solutions of nonlinear problems involving the square root of the Laplacian (2009, background): “7) have been considered by Davila, Dupaigne, and Montenegro [10], [11].”
Semilinear elliptic equations for the fractional Laplacian with Hardy potential (2011, unknown): “α 4 . Now we notice that (γα = mαm−α) (−∆)sϑα = F(F((−∆)sϑα)) = F(ρ2sF(ϑα)(ρ)) = mαF(ρ −N 2 +s−α) = γ αr −2sϑ α. For the proof of the fact that the map α→ γα is continuous and decreasing, we refer to [17]. We deﬁne Υα(t,x) = P(t,·) ∗ ϑα(t,x) ∀(t,x) ∈ RN+1 + ϑα(x) ∀x∈ ∂RN+1 + \ {0}, where P is the”
Regularity of Radial Extremal Solutions for Some Non-Local Semilinear Equations (2010, methodology): “Integrating this inequality with respect to t ∈ [6, 13], recalling that z ≤ Cφ1 for some C > 0, and using (5.”
Regularity of minimizers of semilinear elliptic problems up to dimension 4 (2009, background): “Let u∗ be the extremal solution of (1.5λ).”