On the Elliptic Equations Δu = K(x)u σ and Δu = K(x)e 2u

Conformal metrics with prescribed Gaussian curvature on 𝑆²

“…We omit the details. D We now make an important remark; namely that Lemmas 4.2,4,4,and 4.5 have analogous statements when Kx is replaced by K2. The statements (i) still hold as stated, but e.g.…”

“…Note that / may be empty; for example, if K(r) < 0 for all r > 0, and K(r) < -C/r2 for large r, where C > 0, then 1 = 0 (cf. Sattinger [9], Ni [8], or Cheng and Lin [4]). …”

“…D Using arguments similar to those in the proof of Theorem 2.6, we have, for 0 < r < S and a > a0 > 0, 2" + O (e-^l^a2) . In order to prove this theorem we need two lemmas, whose proofs we omit as they are similar to results of Cheng and Lin [4]. Proof of Theorem 2.12.…”

Existence and nonexistence of blow-up boundary solutions for sublinear elliptic equations

“…the existence, uniqueness and blow-up rate of solutions for problem (1) have been intensively studied in the past few years (see [2][3][4][5][6][7][8][9]11,[13][14][15][16][17], etc.) For the sublinear case f (u) = u m , m ∈ (0, 1), little is known relatively.…”

Conformal metrics with prescribed nonpositive Gaussian curvature on ${\mathbb R}^2$

“…He showed that in case K ≡ −1, equation (1.1) does not have any solution in R 2 . His result was later improved by Sattinger [12], Ni [10] and Cheng and Lin [2]. These results basically contains the following: If K(x) ≤ 0 and K(x) ≤ −|x| −2 near ∞, then equation (1.1) possesses no solutions in R 2 .…”