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By Turner J., Kautz W.H.

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N Let w(v) = v n−2 for v4j +1 ≤ v ≤ v4j , where v4j +1 ∈ (0, v4j ) will be specified momentarily. 97) is v0 t(v) = 1 + d v¯ , w(v) ¯ v4j +1 ≤ v ≤ v0 v and hence v4j t(v4j +1 ) = t(v4j ) + d v¯ n v¯ n−2 v4j +1 ⎞ ⎛ n−2⎝ 1 1 ⎠ . 98) Thus by choosing v4j +1 ∈ (0, v4j /2) sufficiently small and letting t4j +1 = t(v4j +1 ) we have t4j +1 > 4j + 1 and t4j +1 ≤ (n − 2)(1 + ε) 1 . 2 2 v n−2 4j +1 44 Isolated singularities of nonlinear elliptic inequalities Hence v(t4j +1 ) ≤ (n − 2)(1 + ε) 2 n−2 2 1 n−2 2 .

Since our arguments depend on the dimension n, we split our study into the cases n ≥ 3 and n = 2. 1 Optimal a priori bounds The first result of this section reads as follows. 1) in a punctured neighborhood of the origin in Rn (n ≥ 3), where f : (0, ∞) → (0, ∞) is a continuous function satisfying f (t) = O(t n/(n−2) ) 22 as t → ∞. 3) and sup u(x) < C inf u(x) |x|=r |x|=r where C is a constant independent of r. Moreover, if up is summable in some neighborhood of the origin for some p > n/(n − 2) then u has a C 1 extension to the origin and u(0) > 0.

7) under weak regularity assumptions recently appeared in [48]. 5) ranging over the larger set |α| ≤ 2m − 2. 9) ensures that the integral there is finite. 6) as we shall see in Chapter 6. 1) can be found in [19]. 1) in the sense that u(y)(− )m ϕ(y)dy = Rn ϕ(y)dμ(y) Rn The following result appears in [19]. for all ϕ ∈ C0∞ (Rn ). e. x ∈ Rn . 1), essinf u = and u is weakly polysuperharmonic, that is, u(y)(− )i ϕ(y)dy ≥ 0 for all ϕ ∈ C0∞ (Rn ), ϕ ≥ 0, and i = 0, 1, . . , m. e. x ∈ Rn . e. e. x ∈ Rn .

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A survey of progress in graph theory in the Soviet Union by Turner J., Kautz W.H.


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